17039:
17787:
17303:
15020:
3128:
3864:
12388:
6652:
4594:
2869:
12850:
12739:
5121:
can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if
7070:
588:
2295:
12997:
12559:
2422:
3682:
7334:
1784:
1382:
13745:
12244:
6499:
4472:
3123:{\displaystyle g(x)=g(\alpha _{1}\mathbf {e} _{1}+\dots +\alpha _{n}\mathbf {e} _{n})=\alpha _{1}g(\mathbf {e} _{1})+\dots +\alpha _{n}g(\mathbf {e} _{n})=\mathbf {e} ^{1}(x)g(\mathbf {e} _{1})+\dots +\mathbf {e} ^{n}(x)g(\mathbf {e} _{n})}
2600:
12773:
10156:
15346:
12660:
1988:
8069:
2064:
14792:
14213:
6103:
3400:
15554:
8327:
6945:
9813:
6863:
3956:
1486:
453:
7662:
982:
2685:
13246:
12072:
7956:
15206:
11017:
3187:
2797:
1656:
1600:
1233:
1126:
7411:
6357:
2171:
14015:
5865:
14332:
9742:
8183:
4382:
921:
12496:
7848:
3859:{\displaystyle {\begin{bmatrix}e^{11}&e^{12}\\e^{21}&e^{22}\end{bmatrix}}{\begin{bmatrix}e_{11}&e_{21}\\e_{12}&e_{22}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}.}
14526:
7484:
3674:
3621:
3568:
3515:
12888:
9947:
2300:
458:
6451:
8757:
7572:
8386:
1880:
2115:
2166:
13816:
4629:
13866:
11606:
8482:
15990:
8517:
5775:
5525:
10361:
9678:
8597:
7753:
4126:
11413:
10287:
2464:
14950:
8431:
4067:
630:
7154:
5999:
5740:
5624:
5494:
5362:
5304:
5251:
1663:
1261:
11791:
11728:
11668:
10749:
13633:
12383:{\displaystyle {\text{ for all }}A\in {\mathcal {A}}\qquad \|\varphi _{i}-\varphi \|_{A}=\sup _{x\in A}|\varphi _{i}(x)-\varphi (x)|{\underset {i\to \infty }{\longrightarrow }}0.}
6647:{\displaystyle T\left(\sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }\right)=\sum _{\alpha \in A}f_{\alpha }T(e_{\alpha })=\sum _{\alpha \in A}f_{\alpha }\theta _{\alpha }.}
5677:
12653:
12623:
12593:
12489:
8106:
11363:
10964:
8949:
8794:
8559:
5416:
4464:
13896:
9996:
5445:
4743:
4667:
4178:
4014:
3985:
3458:
3429:
3298:
2714:
12162:
11988:
11755:
10503:
9626:
8832:
1521:
349:
13425:
13329:
13084:
12766:
12415:
11917:
11696:
11636:
6409:
2508:
11054:
15781:
15702:
12191:
11435:
6025:
5388:
5330:
4589:{\displaystyle \mathbf {x} =\sum _{i}\langle \mathbf {x} ,\mathbf {e} ^{i}\rangle \mathbf {e} _{i}=\sum _{i}\langle \mathbf {x} ,\mathbf {e} _{i}\rangle \mathbf {e} ^{i},}
2741:
872:
6745:
6682:
5925:
4319:
17823:
16289:
9392:
2860:
1418:
15659:
11437:) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space
4994:
4919:
4873:
396:
288:
15228:
13941:
12211:
12095:
10617:
10444:
9573:
9515:
8889:
5957:
1912:
752:
15877:
15830:
15801:
13579:
13543:
13507:
13474:
13381:
13281:
13165:
13036:
12880:
12236:
11873:
11570:
11487:
6702:
5595:
5562:
313:
10931:
9456:
8977:
6172:
684:
657:
16316:
15904:
15729:
11462:
11158:
10897:
10870:
10843:
9483:
9359:
9332:
8861:
8617:
8217:
7792:
6893:
6222:
5707:
5171:
5119:
5072:
4412:
3241:
3214:
2824:
1814:
1173:
1047:
780:
714:
423:
261:
12845:{\displaystyle {\text{ for all }}A\in {\mathcal {A}}\quad {\text{ and all }}\lambda \in {\mathbb {F} }\quad {\text{ such that }}\lambda \cdot A\in {\mathcal {A}}.}
11191:
4948:
11305:
10645:
10581:
2516:
11079:
9430:
9061:
4244:
95:
16030:
16010:
15851:
15750:
13445:
13401:
13349:
13305:
13128:
13104:
13060:
12459:
12439:
12135:
12115:
11961:
11941:
11893:
11533:
11265:
11215:
11127:
11103:
10816:
10796:
10772:
10689:
10665:
10550:
10530:
10467:
10404:
10384:
10219:
10199:
10179:
10056:
10036:
10016:
9876:
9856:
9836:
9698:
9535:
9305:
9279:
9259:
8911:
8645:
8237:
7142:
7122:
6937:
6917:
6789:
6769:
6491:
6471:
6377:
6306:
6286:
6266:
6246:
6195:
6146:
6126:
6045:
5885:
5818:
5798:
5465:
5271:
5219:
5191:
5140:
5092:
5037:
5014:
4968:
4893:
4847:
4827:
4807:
4787:
4767:
4714:
4687:
4436:
4149:
3269:
2620:
1253:
1146:
1067:
1020:
823:
801:
444:
231:
208:
58:
14104:
12734:{\displaystyle {\text{ for all }}A,B\in {\mathcal {A}}\quad {\text{ there exists some }}C\in {\mathcal {A}}\quad {\text{ such that }}A\cup B\subseteq C.}
10064:
17676:
15663:
1917:
8006:
11794:
1993:
15961:
for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that
15262:
11501:, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of
17816:
17512:
1420:. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations
17339:
14726:
7065:{\displaystyle V^{*}\cong \left(\bigoplus _{\alpha \in A}F\right)^{*}\cong \prod _{\alpha \in A}F^{*}\cong \prod _{\alpha \in A}F\cong F^{A}}
4599:
even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product
6053:
3303:
8242:
583:{\displaystyle {\begin{aligned}(\varphi +\psi )(x)&=\varphi (x)+\psi (x)\\(a\varphi )(x)&=a\left(\varphi (x)\right)\end{aligned}}}
9750:
9229:
is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on
6797:
14151:
8651:. Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.
3872:
1426:
17809:
17502:
11612:
8647:
is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a
132:
vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe
7609:
930:
15497:
2625:
13175:
2290:{\displaystyle x+\lambda y=(\alpha _{1}+\lambda \beta _{1})\mathbf {e} _{1}+\dots +(\alpha _{n}+\lambda \beta _{n})\mathbf {e} _{n}}
17629:
17484:
16897:
15584:
12000:
7861:
16249:
15083:
10969:
5094:, because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of
3136:
2746:
1605:
1549:
1182:
1075:
17460:
15419:
7346:
6311:
3676:. (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as
17230:
13956:
17288:
13039:
11920:
12554:{\displaystyle {\text{ for all }}x\in V\quad {\text{ there exists some }}A\in {\mathcal {A}}\quad {\text{ such that }}x\in A.}
7094:) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be
16832:
16802:
16747:
16605:
16564:
16538:
16452:
5823:
12992:{\displaystyle U_{A}~=~\left\{\varphi \in V'~:~\quad \|\varphi \|_{A}<1\right\},\qquad {\text{ for }}A\in {\mathcal {A}}}
9703:
8128:
4327:
2417:{\displaystyle \mathbf {e} ^{i}(x+\lambda y)=\alpha _{i}+\lambda \beta _{i}=\mathbf {e} ^{i}(x)+\lambda \mathbf {e} ^{i}(y)}
879:
16739:
16711:
11230:, where the dual space has inverse units. Under the natural pairing, these units cancel, and the resulting scalar value is
7814:
15443:
Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual
7416:
3626:
3573:
3520:
3467:
17886:
9884:
6414:
17352:
16681:
13513:, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called
8713:
7539:
15235:
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator
8332:
1823:
17441:
17332:
16719:
16662:
16631:
16512:
16474:
14464:
11825:
2069:
17:
11317:: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to
2120:
17953:
17936:
17711:
17278:
16597:
16504:
16444:
14588:
13010:
8866:
5368:
of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers
4602:
13827:
11575:
9582:
The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space:
8436:
17356:
17240:
17176:
15964:
10553:
8487:
5749:
5499:
10298:
9631:
8564:
7673:
4072:
105:
The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the
17926:
16193:
13604:
13545:
is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of
11388:
11382:
10227:
7997:
7329:{\displaystyle \mathrm {dim} (V)=|A|<|F|^{|A|}=|V^{\ast }|=\mathrm {max} (|\mathrm {dim} (V^{\ast })|,|F|),}
2427:
1779:{\displaystyle \mathbf {e} ^{i}(c^{1}\mathbf {e} _{1}+\cdots +c^{n}\mathbf {e} _{n})=c^{i},\quad i=1,\ldots ,n}
1377:{\displaystyle \mathbf {e} ^{i}(c^{1}\mathbf {e} _{1}+\cdots +c^{n}\mathbf {e} _{n})=c^{i},\quad i=1,\ldots ,n}
13777:
8391:
7491:
4019:
17507:
16794:
16556:
16530:
15604:
14275:
13740:{\displaystyle \|\mathbf {a} \|_{p}=\left(\sum _{n=0}^{\infty }|a_{n}|^{p}\right)^{\frac {1}{p}}<\infty .}
7487:
596:
13589:: the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.
9074:
with its dual. This identity characterizes the transpose, and is formally similar to the definition of the
5962:
5716:
5600:
5470:
5338:
5280:
5227:
18060:
17790:
17563:
17497:
17325:
17018:
16890:
14085:
10697:
114:
14419:
5632:
18075:
17527:
17123:
16973:
14905:
12628:
12598:
12568:
12464:
11239:
8077:
7076:
17981:
17772:
17726:
17650:
17532:
17028:
16922:
15572:
11671:
11320:
10936:
8921:
8766:
8522:
5393:
4441:
137:
124:
Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in
13872:
11764:
11701:
11641:
9955:
5421:
4719:
4643:
4154:
3990:
3961:
3434:
3405:
3274:
2690:
18070:
18002:
17767:
17583:
17268:
16917:
16738:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
16710:. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY:
14116:
12140:
11966:
11733:
10584:
10472:
9585:
9283:
8801:
8660:
5193:, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.
1494:
322:
14038:
13406:
13310:
13065:
12747:
12396:
11898:
11677:
11617:
6382:
2469:
17986:
17958:
17619:
17517:
17420:
17260:
17143:
16042:
15397:
14135:
is a continuous linear map between two topological vector spaces, then the (continuous) transpose
11817:
11730:
the space of compactly supported distributions; and the space of rapidly decreasing test functions
11513:
11509:
the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".
11506:
11502:
11378:
11026:
5710:
110:
15756:
15674:
12169:
11418:
6004:
5371:
5309:
2719:
830:
18065:
17716:
17492:
17306:
17235:
17013:
16883:
15352:
14956:
14380:
14107:, the continuous dual of certain spaces of continuous functions can be described using measures.
6710:
6660:
5890:
4249:
1070:
16257:
9364:
2832:
1390:
17931:
17747:
17691:
17655:
17070:
17003:
16993:
16650:
15637:
15024:
11195:-dimensional space, in the sense that its dimensions can be canceled against the dimensions of
9075:
8685:
8194:
5222:
4973:
4898:
4852:
363:
266:
15213:
13926:
12196:
12080:
10590:
10416:
9540:
9488:
8874:
5930:
1885:
737:
17866:
17085:
17080:
17075:
17008:
16953:
15992:
has a basis. It is also needed to show that the dual of an infinite-dimensional vector space
15786:
15599:
13284:
10775:
9576:
9160:
6748:
6687:
5567:
5534:
4746:
133:
10910:
9435:
8956:
6151:
2595:{\displaystyle \lambda _{1}\mathbf {e} ^{1}+\cdots +\lambda _{n}\mathbf {e} ^{n}=0\in V^{*}}
663:
636:
17965:
17730:
17095:
17060:
17047:
16938:
16820:
16574:
16294:
15882:
15707:
14093:
11798:
11440:
11227:
11136:
10907:
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector
10875:
10848:
10821:
9461:
9337:
9310:
9183:
9130:
8839:
8602:
8202:
7777:
7080:
6871:
6200:
5685:
5274:
5149:
5097:
5050:
4390:
3219:
3192:
2802:
1792:
1151:
1025:
758:
692:
401:
239:
17317:
16582:
11167:
9575:. Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the
4924:
8:
18039:
17896:
17871:
17696:
17634:
17348:
17273:
17153:
17128:
16978:
14061:
14056:(the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces
13131:
11282:
10622:
10558:
9134:
8648:
7515:
211:
145:
15857:
15810:
13559:
13523:
13487:
13454:
13361:
13261:
13145:
13016:
12860:
12216:
11853:
11550:
11467:
11061:
9397:
8986:
4183:
293:
77:
17916:
17721:
17588:
16983:
16786:
16774:
16522:
16015:
15995:
15836:
15735:
15614:
15609:
13430:
13386:
13334:
13290:
13113:
13089:
13045:
12444:
12424:
12120:
12100:
11946:
11926:
11878:
11518:
11250:
11220:
11200:
11112:
11088:
10801:
10781:
10757:
10674:
10650:
10535:
10515:
10452:
10389:
10369:
10204:
10184:
10164:
10041:
10021:
10001:
9861:
9841:
9821:
9683:
9520:
9290:
9264:
9244:
8896:
8630:
8222:
7759:
7127:
7107:
6922:
6902:
6774:
6754:
6476:
6456:
6362:
6291:
6271:
6251:
6231:
6180:
6131:
6111:
6030:
5870:
5803:
5783:
5743:
5450:
5256:
5204:
5176:
5125:
5077:
5040:
5022:
4999:
4953:
4878:
4832:
4812:
4792:
4772:
4752:
4699:
4672:
4421:
4134:
3254:
2605:
1238:
1131:
1052:
1005:
808:
786:
429:
216:
193:
129:
43:
13167:
is normed (in fact a Banach space if the field of scalars is complete), with the norm
17948:
17701:
17181:
17138:
17065:
16958:
16861:
16858:
16838:
16828:
16808:
16798:
16753:
16743:
16733:
16715:
16704:
16687:
16677:
16658:
16642:
16627:
16615:
16601:
16560:
16534:
16508:
16470:
16448:
16189:
15928:
15920:
14455:
14097:
10506:
7985:
353:
17801:
14120:
18018:
17706:
17624:
17593:
17573:
17558:
17553:
17548:
17186:
17090:
16943:
16797:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
16619:
16578:
16484:
16462:
16059:
14401:
13546:
13255:
11850:
There is a standard construction for introducing a topology on the continuous dual
11611:
Important examples for continuous dual spaces are the space of compactly supported
11235:
10668:
10151:{\displaystyle \left(\bigcup _{i\in I}A_{i}\right)^{0}=\bigcap _{i\in I}A_{i}^{0}.}
9233:, which identifies the space of column vectors with the dual space of row vectors.
9159:, taking the dual of vector spaces and the transpose of linear maps is therefore a
9138:
7145:
925:
176:
17385:
11019:
to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to
17943:
17836:
17568:
17522:
17470:
17465:
17436:
17245:
17038:
16998:
16988:
16570:
16045:
with domain a vector space and the space of functionals on the dual vector space.
15958:
15427:
15386:
14992:
13510:
13477:
11814:
11494:
10407:
9156:
7981:
7091:
4415:
1524:
985:
17395:
7086:
If a vector space is not finite-dimensional, then its (algebraic) dual space is
1983:{\displaystyle x=\alpha _{1}\mathbf {e} _{1}+\dots +\alpha _{n}\mathbf {e} _{n}}
17881:
17757:
17609:
17410:
17250:
17171:
16906:
15422:, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if
14089:
11841:
11758:
11366:
11314:
8064:{\displaystyle \Phi _{\langle \cdot ,\cdot \rangle }:V\to {\overline {V^{*}}}.}
7340:
6896:
3869:
Solving for the unknown values in the first matrix shows the dual basis to be
2059:{\displaystyle y=\beta _{1}\mathbf {e} _{1}+\dots +\beta _{n}\mathbf {e} _{n}}
18054:
17921:
17904:
17861:
17762:
17686:
17415:
17400:
17390:
17283:
17206:
17166:
17133:
17113:
16842:
16812:
16691:
16676:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
16646:
16436:
15341:{\displaystyle \Psi (x)(\varphi )=\varphi (x),\quad x\in V,\ \varphi \in V'.}
14065:
13582:
13355:
13139:
11231:
11020:
10469:
with its image in the second dual space under the double duality isomorphism
7519:
141:
16757:
7101:
The proof of this inequality between dimensions results from the following.
4631:
and the corresponding duality pairing are introduced, as described below in
17752:
17405:
17375:
17216:
17105:
17055:
16948:
16729:
16699:
16589:
16328:
16326:
15571:, and continuity of Ψ depends upon this choice. As a consequence, defining
15440:
of the continuous dual, again as a consequence of the Hahn–Banach theorem.
15046:
14088:, the continuous dual of a Hilbert space is again a Hilbert space which is
13135:
11243:
357:
188:
37:
11365:. Similarly, if the primal space measures length, the dual space measures
8108:
can be identified with the set of all additive complex-valued functionals
17856:
17832:
17681:
17671:
17578:
17380:
17196:
17161:
17118:
16963:
16496:
16245:
15589:
11845:
9112:
8624:
7098:
to the original vector space even if the latter is infinite-dimensional.
4689:
726:
70:
33:
16323:
4438:. The biorthogonality property of these two basis sets allows any point
17614:
16968:
16548:
14831:
14787:{\displaystyle W^{\perp }=\{\varphi \in V':W\subseteq \ker \varphi \}.}
14241:
produces a linear map between the space of continuous linear maps from
12857:
If these requirements are fulfilled then the corresponding topology on
8680:
8197:
7095:
5143:
1176:
997:
316:
16081:
16079:
14075:(the sequences converging to zero) are both naturally identified with
5780:
This observation generalizes to any infinite-dimensional vector space
5273:, then the same construction as in the finite-dimensional case yields
18023:
17909:
17876:
17023:
16866:
16559:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
15704:
is reserved for some other meaning. For instance, in the above text,
15594:
15382:
15019:
13586:
11270:
9202:
9092:
9070:
with its dual space, and that on the right is the natural pairing of
8620:
6174:
may be written uniquely in this way by the definition of the basis).
6098:{\displaystyle \sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }}
5044:
3395:{\displaystyle \{\mathbf {e} _{1}=(1/2,1/2),\mathbf {e} _{2}=(0,1)\}}
447:
when equipped with an addition and scalar multiplication satisfying:
99:
13903:-th term is 1 and all others are zero. Conversely, given an element
9628:, and the annihilator of the whole space is just the zero covector:
8322:{\displaystyle V^{**}=\{\Phi :V^{*}\to F:\Phi \ \mathrm {linear} \}}
7810:
is finite dimensional) defines a unique nondegenerate bilinear form
17191:
16076:
15356:
13609:
13585:, then the corresponding reflexivity is presented in the theory of
11991:
9808:{\displaystyle \{0\}\subseteq T^{0}\subseteq S^{0}\subseteq V^{*}.}
7988:
instead of bilinear forms. In that case, a given sesquilinear form
6858:{\displaystyle V\cong (F^{A})_{0}\cong \bigoplus _{\alpha \in A}F.}
5682:
which is a finite sum because there are only finitely many nonzero
5365:
3461:
15079:. The addition +′ induced by the transformation can be defined as
14208:{\displaystyle T'(\varphi )=\varphi \circ T,\quad \varphi \in W'.}
7961:
Thus there is a one-to-one correspondence between isomorphisms of
3951:{\displaystyle \{\mathbf {e} ^{1}=(2,0),\mathbf {e} ^{2}=(-1,1)\}}
1481:{\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}}
16875:
14561:
are equipped with "compatible" topologies: for example, when for
14535:
is a continuous linear map between two topological vector spaces
7603:. In other words, the bilinear form determines a linear mapping
16041:
To be precise, continuous
Fourier analysis studies the space of
10509:
on the lattice of subsets of a finite-dimensional vector space.
7657:{\displaystyle \Phi _{\langle \cdot ,\cdot \rangle }:V\to V^{*}}
6771:(viewed as a 1-dimensional vector space over itself) indexed by
977:{\displaystyle \langle \cdot ,\cdot \rangle :V\times V^{*}\to F}
17201:
15549:{\displaystyle \varphi \in V'\mapsto \varphi (x),\quad x\in V,}
11674:(generalized functions); the space of arbitrary test functions
11310:
10583:
is a vector space in its own right, and so has a dual. By the
9680:. Furthermore, the assignment of an annihilator to a subset of
7770:
is finite-dimensional, then this is an isomorphism onto all of
2680:{\displaystyle \lambda _{1}=\lambda _{2}=\dots =\lambda _{n}=0}
125:
13241:{\displaystyle \|\varphi \|=\sup _{\|x\|\leq 1}|\varphi (x)|.}
15575:
in this framework is more involved than in the normed case.
13331:
can be chosen as the class of all totally bounded subsets in
12067:{\displaystyle \|\varphi \|_{A}=\sup _{x\in A}|\varphi (x)|,}
11572:
is defined as the space of all continuous linear functionals
7951:{\displaystyle \langle v,w\rangle _{\Phi }=(\Phi (v))(w)=.\,}
15201:{\displaystyle (\varphi )=\varphi (x_{1}+x_{2})=\varphi (x)}
13383:
is the topology of uniform convergence on finite subsets in
12768:
is closed under the operation of multiplication by scalars:
11012:{\displaystyle \langle x,\varphi \rangle :=\varphi (x)\in F}
3402:. The basis vectors are not orthogonal to each other. Then,
3182:{\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}}
2792:{\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}}
1651:{\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}}
1595:{\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}
1228:{\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}}
1121:{\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}}
17347:
15027:
of vector addition from a vector space to its double dual.
7406:{\displaystyle \mathrm {dim} (V)<\mathrm {dim} (V^{*}),}
6352:{\displaystyle \theta _{\alpha }=T(\mathbf {e} _{\alpha })}
4321:
is a matrix whose columns are the dual basis vectors, then
16856:
16401:
16377:
15455:
is not uniquely defined as a set. Saying that Ψ maps from
14010:{\displaystyle \varphi (\mathbf {b} )=\sum _{n}a_{n}b_{n}}
11547:(in the sense of the theory of topological vector spaces)
7577:
where the right hand side is defined as the functional on
144:. Consequently, the dual space is an important concept in
113:, there is a subspace of the dual space, corresponding to
16343:
16341:
15485:, is a reasonable minimal requirement on the topology of
14999:
itself. The converse is not true: for example, the space
11493:
normed vector space or topological vector space, such as
9066:
where the bracket on the left is the natural pairing of
16389:
14429:
is a
Hilbert space, there is an antilinear isomorphism
14110:
6747:
may be identified (essentially by definition) with the
5860:{\displaystyle \{\mathbf {e} _{\alpha }:\alpha \in A\}}
5447:
is the sequence consisting of all zeroes except in the
16338:
16250:"A mathematical formalisation of dimensional analysis"
15404:) can still be defined by the same formula, for every
14379:. Several properties of transposition depend upon the
14096:
used by physicists in the mathematical formulation of
13602:< ∞ be a real number and consider the Banach space
11385:
linear functionals from the space into the base field
9737:{\displaystyle \{0\}\subseteq S\subseteq T\subseteq V}
9095:
linear map between the space of linear operators from
8178:{\displaystyle f(\alpha v)={\overline {\alpha }}f(v).}
4692:, its dual space is typically written as the space of
4633:
4377:{\displaystyle {\hat {E}}^{\textrm {T}}\cdot E=I_{n},}
3822:
3755:
3691:
916:{\displaystyle \varphi (x)=\langle x,\varphi \rangle }
17831:
16297:
16260:
16018:
15998:
15967:
15885:
15860:
15839:
15813:
15789:
15759:
15738:
15710:
15677:
15640:
15500:
15265:
15216:
15086:
14908:
14729:
14467:
14278:
14154:
13959:
13929:
13875:
13830:
13780:
13636:
13562:
13526:
13490:
13457:
13433:
13409:
13389:
13364:
13337:
13313:
13293:
13264:
13178:
13148:
13116:
13092:
13086:
can be chosen as the class of all bounded subsets in
13068:
13048:
13019:
12891:
12863:
12776:
12750:
12663:
12631:
12601:
12571:
12499:
12467:
12447:
12427:
12399:
12247:
12219:
12199:
12172:
12143:
12123:
12103:
12083:
12003:
11990:
or what is the same thing, the topology generated by
11969:
11949:
11929:
11901:
11881:
11856:
11767:
11736:
11704:
11680:
11644:
11620:
11578:
11553:
11521:
11470:
11443:
11421:
11391:
11323:
11285:
11253:
11203:
11170:
11139:
11115:
11091:
11064:
11029:
10972:
10939:
10913:
10878:
10851:
10824:
10804:
10784:
10760:
10700:
10677:
10653:
10625:
10593:
10561:
10538:
10518:
10475:
10455:
10419:
10392:
10372:
10301:
10230:
10207:
10187:
10167:
10067:
10044:
10024:
10004:
9958:
9887:
9864:
9844:
9824:
9753:
9706:
9686:
9634:
9588:
9543:
9523:
9491:
9464:
9438:
9400:
9367:
9340:
9313:
9293:
9267:
9247:
8989:
8959:
8924:
8899:
8877:
8842:
8804:
8769:
8716:
8633:
8605:
8567:
8525:
8490:
8439:
8394:
8335:
8245:
8225:
8205:
8131:
8080:
8009:
7984:
field, then sometimes it is more natural to consider
7864:
7843:{\displaystyle \langle \cdot ,\cdot \rangle _{\Phi }}
7817:
7780:
7676:
7612:
7542:
7419:
7349:
7157:
7130:
7110:
6948:
6925:
6905:
6874:
6800:
6777:
6757:
6713:
6690:
6663:
6502:
6479:
6459:
6417:
6385:
6365:
6314:
6294:
6274:
6254:
6234:
6203:
6183:
6154:
6134:
6114:
6056:
6033:
6007:
5965:
5933:
5893:
5873:
5826:
5806:
5786:
5752:
5719:
5688:
5635:
5603:
5570:
5537:
5502:
5473:
5453:
5424:
5396:
5374:
5341:
5332:) of the dual space, but they will not form a basis.
5312:
5283:
5259:
5230:
5207:
5179:
5152:
5128:
5100:
5080:
5053:
5025:
5002:
4976:
4956:
4927:
4901:
4881:
4855:
4835:
4815:
4795:
4775:
4755:
4722:
4702:
4675:
4646:
4605:
4475:
4444:
4424:
4393:
4330:
4252:
4186:
4157:
4137:
4075:
4022:
3993:
3964:
3875:
3685:
3629:
3576:
3523:
3470:
3437:
3408:
3306:
3277:
3257:
3222:
3195:
3139:
2872:
2835:
2805:
2749:
2722:
2693:
2628:
2608:
2519:
2472:
2430:
2303:
2174:
2123:
2072:
1996:
1920:
1888:
1826:
1795:
1666:
1608:
1552:
1497:
1429:
1393:
1264:
1241:
1185:
1154:
1134:
1078:
1055:
1028:
1008:
933:
882:
833:
811:
789:
761:
740:
695:
666:
639:
599:
456:
432:
404:
366:
325:
296:
269:
242:
219:
196:
80:
46:
16825:
Topological Vector Spaces, Distributions and
Kernels
16521:
16097:
16062:
of Ψ is the smallest closed subspace containing {0}.
15671:, p. 19). This notation is sometimes used when
15565:. Further, there is still a choice of a topology on
13427:
can be chosen as the class of all finite subsets in
7497:
7479:{\displaystyle |F|\leq |\mathrm {dim} (V^{\ast })|,}
4246:
is a matrix whose columns are the basis vectors and
3669:{\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{2})=1}
3616:{\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{1})=0}
3563:{\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{2})=0}
3510:{\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{1})=1}
3464:(functions that map a vector to a scalar) such that
16641:
16171:
16160:
15731:is frequently used to denote the codifferential of
14706:
is a closed linear subspace of a normed space
9942:{\displaystyle A^{0}+B^{0}\subseteq (A\cap B)^{0}.}
9236:
4921:matrix (trivially, a real number) respectively, if
2602:. Applying this functional on the basis vectors of
17677:Spectral theory of ordinary differential equations
16764:
16703:
16489:Elements of mathematics, Topological vector spaces
16332:
16310:
16283:
16186:Elements of mathematics: Algebra I, Chapters 1 - 3
16024:
16004:
15984:
15898:
15871:
15845:
15824:
15795:
15775:
15744:
15723:
15696:
15653:
15548:
15414:, however several difficulties arise. First, when
15340:
15222:
15200:
14944:
14786:
14520:
14326:
14207:
14009:
13935:
13890:
13860:
13810:
13739:
13573:
13537:
13501:
13468:
13439:
13419:
13395:
13375:
13343:
13323:
13299:
13275:
13240:
13159:
13122:
13098:
13078:
13054:
13030:
12991:
12874:
12844:
12760:
12733:
12647:
12617:
12587:
12553:
12483:
12453:
12433:
12409:
12382:
12230:
12205:
12185:
12156:
12129:
12109:
12089:
12066:
11982:
11955:
11935:
11911:
11887:
11867:
11785:
11749:
11722:
11690:
11662:
11630:
11600:
11564:
11527:
11481:
11456:
11429:
11407:
11357:
11299:
11259:
11209:
11185:
11152:
11121:
11097:
11073:
11048:
11011:
10958:
10925:
10891:
10864:
10837:
10810:
10790:
10766:
10743:
10683:
10659:
10639:
10611:
10575:
10544:
10524:
10497:
10461:
10438:
10398:
10378:
10355:
10281:
10213:
10193:
10173:
10150:
10050:
10030:
10010:
9990:
9941:
9870:
9850:
9830:
9807:
9736:
9692:
9672:
9620:
9567:
9529:
9509:
9477:
9450:
9424:
9386:
9353:
9326:
9299:
9273:
9253:
9175:using the natural injection into the double dual.
9055:
8971:
8943:
8905:
8883:
8855:
8826:
8788:
8751:
8639:
8611:
8591:
8553:
8511:
8476:
8425:
8380:
8321:
8231:
8211:
8177:
8100:
8063:
7950:
7842:
7786:
7747:
7656:
7566:
7490:. The exact dimension of the dual is given by the
7478:
7405:
7328:
7136:
7116:
7064:
6931:
6911:
6887:
6857:
6783:
6763:
6739:
6696:
6676:
6646:
6485:
6465:
6446:{\displaystyle \theta (\alpha )=\theta _{\alpha }}
6445:
6403:
6371:
6351:
6300:
6280:
6260:
6240:
6216:
6189:
6166:
6140:
6120:
6097:
6039:
6019:
5993:
5951:
5919:
5879:
5859:
5812:
5792:
5769:
5734:
5701:
5671:
5618:
5589:
5556:
5519:
5488:
5459:
5439:
5410:
5382:
5356:
5324:
5298:
5265:
5245:
5213:
5185:
5165:
5134:
5113:
5086:
5066:
5031:
5008:
4988:
4962:
4942:
4913:
4887:
4867:
4841:
4821:
4801:
4781:
4761:
4737:
4708:
4681:
4661:
4623:
4588:
4458:
4430:
4406:
4376:
4313:
4238:
4172:
4143:
4120:
4061:
4008:
3979:
3950:
3858:
3668:
3615:
3562:
3509:
3452:
3423:
3394:
3292:
3263:
3235:
3208:
3181:
3122:
2854:
2818:
2791:
2735:
2708:
2679:
2614:
2594:
2502:
2458:
2416:
2289:
2160:
2109:
2058:
1982:
1906:
1874:
1808:
1778:
1650:
1594:
1515:
1480:
1412:
1376:
1247:
1227:
1167:
1148:, it is possible to construct a specific basis in
1140:
1120:
1061:
1041:
1014:
976:
915:
866:
817:
795:
774:
746:
708:
678:
651:
624:
582:
438:
417:
390:
343:
307:
282:
255:
225:
202:
89:
52:
16671:
16085:
13922:, the corresponding continuous linear functional
13005:Here are the three most important special cases.
12417:is supposed to satisfy the following conditions:
11824:is identical to the continuous dual space of the
8752:{\displaystyle f^{*}(\varphi )=\varphi \circ f\,}
8188:
7762:, then this is an isomorphism onto a subspace of
7567:{\displaystyle v\mapsto \langle v,\cdot \rangle }
102:addition and scalar multiplication by constants.
18052:
16614:
16208:
16183:
16012:is nonzero, and hence that the natural map from
15430:and locally convex, the map Ψ is injective from
13549:: the spaces reflexive in this sense are called
13192:
12300:
12024:
11797:(slowly growing distributions) in the theory of
8381:{\displaystyle (\Psi (v))(\varphi )=\varphi (v)}
1875:{\displaystyle \mathbf {e} ^{i},i=1,2,\dots ,n,}
15957:Several assertions in this article require the
14521:{\displaystyle i_{V}\circ T^{*}=T'\circ i_{V}.}
14345:are normed spaces, the norm of the transpose in
10505:. In particular, forming the annihilator is a
2110:{\displaystyle \mathbf {e} ^{i}(x)=\alpha _{i}}
16672:Narici, Lawrence; Beckenstein, Edward (2011).
14092:to the original space. This gives rise to the
11242:: given a one-dimensional vector space with a
7486:which can be done with an argument similar to
7144:-vector space, the arithmetical properties of
5531:sequences of real numbers: each real sequence
4669:can be interpreted as the space of columns of
2161:{\displaystyle \mathbf {e} ^{i}(y)=\beta _{i}}
17817:
17333:
16891:
16785:
14387:has dense range if and only if the transpose
9129:then the space of linear maps is actually an
5142:is a vector space of any dimension, then the
4624:{\displaystyle \langle \cdot ,\cdot \rangle }
14778:
14743:
14269:are composable continuous linear maps, then
14105:Riesz–Markov–Kakutani representation theorem
14044:In a similar manner, the continuous dual of
13861:{\displaystyle (\varphi (\mathbf {e} _{n}))}
13646:
13637:
13202:
13196:
13185:
13179:
12947:
12940:
12287:
12267:
12011:
12004:
11601:{\displaystyle \varphi :V\to {\mathbb {F} }}
11234:, as expected. For example, in (continuous)
10985:
10973:
9760:
9754:
9713:
9707:
9654:
9648:
9596:
9589:
8654:
8477:{\displaystyle \mathrm {ev} _{v}:V^{*}\to F}
8316:
8262:
8027:
8015:
7878:
7865:
7831:
7818:
7739:
7727:
7699:
7687:
7630:
7618:
7561:
7549:
5854:
5827:
5467:-th position, which is 1. The dual space of
5196:
4618:
4606:
4568:
4545:
4517:
4494:
3945:
3876:
3389:
3307:
3176:
3140:
2786:
2750:
1645:
1609:
1589:
1553:
1527:symbol. This property is referred to as the
1222:
1186:
1115:
1079:
946:
934:
910:
898:
98:together with the vector space structure of
27:In mathematics, vector space of linear forms
15985:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
13451:Each of these three choices of topology on
8512:{\displaystyle \varphi \mapsto \varphi (v)}
5770:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
5520:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
4809:. Then, seeing this functional as a matrix
17824:
17810:
17454:
17340:
17326:
16898:
16884:
14605:of uniform convergence on bounded sets of
14145:is defined by the same formula as before:
13283:is the topology of uniform convergence on
13038:is the topology of uniform convergence on
10356:{\displaystyle (A\cap B)^{0}=A^{0}+B^{0}.}
9673:{\displaystyle V^{0}=\{0\}\subseteq V^{*}}
9361:, is the collection of linear functionals
8592:{\displaystyle v\mapsto \mathrm {ev} _{v}}
7748:{\displaystyle \left=\langle v,w\rangle .}
4749:. This is because a functional maps every
4121:{\displaystyle \mathbf {e} ^{2}(x,y)=-x+y}
4016:are functionals, they can be rewritten as
991:
17450:
16058:is locally convex but not Hausdorff, the
15976:
15970:
15953:
15951:
15949:
15947:
15879:for the continuous dual, while reserving
15559:be continuous for the chosen topology on
15381:. Normed spaces for which the map Ψ is a
14418:is compact. This can be proved using the
12809:
11835:
11820:(TVS), then the continuous dual space of
11593:
11423:
11408:{\displaystyle \mathbb {F} =\mathbb {C} }
11401:
11393:
10282:{\displaystyle (A+B)^{0}=A^{0}\cap B^{0}}
9034:
9015:
8748:
7947:
5761:
5755:
5722:
5606:
5511:
5505:
5476:
5404:
5376:
5344:
4725:
4649:
4160:
3280:
2459:{\displaystyle \mathbf {e} ^{i}\in V^{*}}
17630:Group algebra of a locally compact group
16773:
16626:(3rd ed.). AMS Chelsea Publishing.
16527:A (Terse) Introduction to Linear Algebra
16483:
16461:
16407:
16395:
16383:
16371:
16347:
16184:Nicolas Bourbaki (1974). Hermann (ed.).
15585:Covariance and contravariance of vectors
15018:
13811:{\displaystyle \varphi \in (\ell ^{p})'}
12393:Usually (but not necessarily) the class
11372:
9167:to itself. It is possible to identify (
9163:from the category of vector spaces over
8426:{\displaystyle v\in V,\varphi \in V^{*}}
6128:(the sum is finite by the assumption on
4634:§ Bilinear products and dual spaces
4062:{\displaystyle \mathbf {e} ^{1}(x,y)=2x}
16765:Robertson, A.P.; Robertson, W. (1964).
14327:{\displaystyle (U\circ T)'=T'\circ U'.}
11309:). For example, if time is measured in
10902:
9103:and the space of linear operators from
924:. This pairing defines a nondegenerate
625:{\displaystyle \varphi ,\psi \in V^{*}}
356:). Since linear maps are vector space
182:
14:
18053:
17289:Comparison of linear algebra libraries
16819:
16495:
16232:
16220:
16148:
16136:
15944:
15907:
15804:
15617:– dual space basis, in crystallography
15491:, namely that the evaluation mappings
14383:. For example, the bounded linear map
9225:acting on the left on column vectors,
6197:may then be identified with the space
5994:{\displaystyle f_{\alpha }=f(\alpha )}
5735:{\displaystyle \mathbb {R} ^{\infty }}
5619:{\displaystyle \mathbb {R} ^{\infty }}
5489:{\displaystyle \mathbb {R} ^{\infty }}
5357:{\displaystyle \mathbb {R} ^{\infty }}
5299:{\displaystyle \mathbf {e} ^{\alpha }}
5246:{\displaystyle \mathbf {e} _{\alpha }}
17805:
17321:
16879:
16857:
16827:. Mineola, N.Y.: Dover Publications.
16728:
16698:
16435:
16419:
16359:
16124:
16108:
16106:
15449:, so that the continuous double dual
15049:of two vectors. The addition + sends
14982:
14404:linear map between two Banach spaces
12166:This means that a net of functionals
12097:is a continuous linear functional on
11313:, the corresponding dual unit is the
10744:{\displaystyle (V/W)^{*}\cong W^{0}.}
8917:The following identity holds for all
6791:, i.e. there are linear isomorphisms
6308:is uniquely determined by the values
5564:defines a function where the element
5039:consists of the space of geometrical
689:Elements of the algebraic dual space
16740:McGraw-Hill Science/Engineering/Math
16712:McGraw-Hill Science/Engineering/Math
16547:
15783:represents the pullback of the form
14816:can be identified with the quotient
14111:Transpose of a continuous linear map
11963:of uniform convergence on sets from
9217:, hence the name. Alternatively, as
7969:and nondegenerate bilinear forms on
5672:{\displaystyle \sum _{n}a_{n}x_{n},}
5221:is not finite-dimensional but has a
17887:Topologies on spaces of linear maps
16244:
14945:{\displaystyle \ker(j')=W^{\perp }}
14889:, then the kernel of the transpose
14609:, or both have the weak-∗ topology
12648:{\displaystyle C\in {\mathcal {A}}}
12618:{\displaystyle B\in {\mathcal {A}}}
12588:{\displaystyle A\in {\mathcal {A}}}
12484:{\displaystyle A\in {\mathcal {A}}}
9485:consists of all linear functionals
8101:{\displaystyle {\overline {V^{*}}}}
7339:where cardinalities are denoted as
5074:form a family of parallel lines in
4745:as a linear functional by ordinary
826:is sometimes denoted by a bracket:
425:itself becomes a vector space over
24:
16905:
16781:. New York: The Macmillan Company.
16588:
16467:Elements of mathematics, Algebra I
16112:
16103:
16098:Katznelson & Katznelson (2008)
15668:
15266:
15117:
15090:
14710:, and consider the annihilator of
14249:and the space of linear maps from
13731:
13680:
13412:
13316:
13071:
12984:
12834:
12790:
12753:
12702:
12683:
12640:
12610:
12580:
12528:
12476:
12402:
12371:
12261:
12146:
11972:
11904:
11771:
11739:
11708:
11683:
11648:
11623:
9209:with respect to the dual bases of
8606:
8579:
8576:
8526:
8445:
8442:
8339:
8312:
8309:
8306:
8303:
8300:
8297:
8290:
8265:
8206:
8011:
7923:
7893:
7882:
7835:
7781:
7683:
7614:
7448:
7445:
7442:
7380:
7377:
7374:
7357:
7354:
7351:
7279:
7276:
7273:
7260:
7257:
7254:
7165:
7162:
7159:
6684:is nonzero for only finitely many
6001:is nonzero for only finitely many
5727:
5611:
5481:
5349:
1882:are linear functionals, which map
25:
18087:
16850:
16525:; Katznelson, Yonatan R. (2008).
16172:Misner, Thorne & Wheeler 1973
16161:Misner, Thorne & Wheeler 1973
14965:induces an isometric isomorphism
14855:is an isometric isomorphism from
14624:of pointwise convergence on
11505:. Nevertheless, in the theory of
11358:{\displaystyle 3s\cdot 2s^{-1}=6}
10959:{\displaystyle \varphi \in V^{*}}
8944:{\displaystyle \varphi \in W^{*}}
8789:{\displaystyle \varphi \in W^{*}}
8554:{\displaystyle \Psi :V\to V^{**}}
8484:is the evaluation map defined by
7965:to a subspace of (resp., all of)
7498:Bilinear products and dual spaces
6657:Again, the sum is finite because
5777:does not have a countable basis.
5411:{\displaystyle i\in \mathbb {N} }
5335:For instance, consider the space
4716:real numbers. Such a row acts on
4459:{\displaystyle \mathbf {x} \in V}
17786:
17785:
17712:Topological quantum field theory
17302:
17301:
17279:Basic Linear Algebra Subprograms
17037:
16501:Finite-Dimensional Vector Spaces
16032:to its double dual is injective.
15832:to denote the algebraic dual of
15250:into its continuous double dual
13967:
13891:{\displaystyle \mathbf {e} _{n}}
13878:
13842:
13641:
11786:{\displaystyle {\mathcal {S}}',}
11723:{\displaystyle {\mathcal {E}}',}
11663:{\displaystyle {\mathcal {D}}',}
10754:As a particular consequence, if
10691:. There is thus an isomorphism
9991:{\displaystyle (A_{i})_{i\in I}}
9700:reverses inclusions, so that if
9237:Quotient spaces and annihilators
9137:, and the assignment is then an
8074:The conjugate of the dual space
6539:
6336:
6085:
5832:
5440:{\displaystyle \mathbf {e} _{i}}
5427:
5286:
5233:
4738:{\displaystyle \mathbb {R} ^{n}}
4662:{\displaystyle \mathbb {R} ^{n}}
4573:
4558:
4549:
4522:
4507:
4498:
4477:
4446:
4298:
4273:
4223:
4198:
4173:{\displaystyle \mathbb {R} ^{n}}
4078:
4025:
4009:{\displaystyle \mathbf {e} ^{2}}
3996:
3980:{\displaystyle \mathbf {e} ^{1}}
3967:
3914:
3881:
3647:
3632:
3594:
3579:
3541:
3526:
3488:
3473:
3453:{\displaystyle \mathbf {e} ^{2}}
3440:
3424:{\displaystyle \mathbf {e} ^{1}}
3411:
3361:
3312:
3293:{\displaystyle \mathbb {R} ^{2}}
3166:
3145:
3107:
3080:
3056:
3029:
3011:
2971:
2937:
2906:
2776:
2755:
2709:{\displaystyle \mathbf {e} _{i}}
2696:
2563:
2532:
2433:
2395:
2368:
2306:
2277:
2224:
2126:
2075:
2046:
2015:
1970:
1939:
1829:
1725:
1694:
1669:
1635:
1614:
1579:
1558:
1447:
1432:
1323:
1292:
1267:
1212:
1191:
1105:
1084:
360:, the dual space may be denoted
17177:Seven-dimensional cross product
16429:
16413:
16365:
16353:
16238:
16226:
16214:
16202:
16177:
16165:
16048:
16035:
15533:
15305:
14881:denotes the injection map from
14797:Then, the dual of the quotient
14697:
14446:. For every bounded linear map
14187:
13818:, the corresponding element of
13480:for topological vector spaces:
12970:
12939:
12814:
12795:
12707:
12688:
12533:
12514:
12266:
12157:{\displaystyle {\mathcal {A}}.}
11983:{\displaystyle {\mathcal {A}},}
11750:{\displaystyle {\mathcal {S}},}
10498:{\displaystyle V\approx V^{**}}
9621:{\displaystyle \{0\}^{0}=V^{*}}
8827:{\displaystyle f^{*}(\varphi )}
7774:. Conversely, any isomorphism
7518:between these two spaces. Any
1754:
1516:{\displaystyle \delta _{j}^{i}}
1387:for any choice of coefficients
1352:
344:{\displaystyle \varphi :V\to F}
315:) is defined as the set of all
16333:Robertson & Robertson 1964
16154:
16142:
16130:
16118:
16091:
15913:
15854:. However, other authors use
15685:
15678:
15628:
15527:
15521:
15515:
15299:
15293:
15284:
15278:
15275:
15269:
15195:
15189:
15180:
15154:
15145:
15139:
15136:
15133:
15120:
15106:
15093:
15087:
15014:
14987:If the dual of a normed space
14926:
14915:
14292:
14279:
14169:
14163:
13971:
13963:
13855:
13852:
13837:
13831:
13801:
13787:
13766:. Then the continuous dual of
13702:
13686:
13420:{\displaystyle {\mathcal {A}}}
13324:{\displaystyle {\mathcal {A}}}
13231:
13227:
13221:
13214:
13142:) then the strong topology on
13079:{\displaystyle {\mathcal {A}}}
12761:{\displaystyle {\mathcal {A}}}
12410:{\displaystyle {\mathcal {A}}}
12368:
12361:
12355:
12351:
12345:
12336:
12330:
12316:
12057:
12053:
12047:
12040:
11912:{\displaystyle {\mathcal {A}}}
11875:of a topological vector space
11691:{\displaystyle {\mathcal {E}}}
11631:{\displaystyle {\mathcal {D}}}
11588:
11269:, the dual space has units of
11180:
11171:
11000:
10994:
10933:can be paired with a covector
10716:
10701:
10603:
10315:
10302:
10244:
10231:
9973:
9959:
9927:
9914:
9549:
9501:
9413:
9401:
9047:
9044:
9038:
9025:
9019:
9009:
9003:
8990:
8821:
8815:
8733:
8727:
8571:
8535:
8506:
8500:
8494:
8468:
8375:
8369:
8360:
8354:
8351:
8348:
8342:
8336:
8281:
8189:Injection into the double-dual
8169:
8163:
8144:
8135:
8038:
7941:
7932:
7926:
7920:
7914:
7908:
7905:
7902:
7896:
7890:
7710:
7704:
7641:
7546:
7469:
7465:
7452:
7437:
7429:
7421:
7397:
7384:
7367:
7361:
7320:
7316:
7308:
7300:
7296:
7283:
7268:
7264:
7246:
7231:
7221:
7213:
7207:
7198:
7190:
7182:
7175:
7169:
6895:is (again by definition), the
6821:
6807:
6728:
6714:
6599:
6586:
6453:) defines a linear functional
6427:
6421:
6404:{\displaystyle \theta :A\to F}
6395:
6346:
6331:
6047:is identified with the vector
5988:
5982:
5943:
5908:
5894:
5584:
5571:
5551:
5538:
4338:
4308:
4292:
4284:
4268:
4259:
4233:
4217:
4209:
4193:
4100:
4088:
4047:
4035:
3942:
3927:
3906:
3894:
3657:
3642:
3604:
3589:
3551:
3536:
3498:
3483:
3386:
3374:
3353:
3325:
3117:
3102:
3096:
3090:
3066:
3051:
3045:
3039:
3021:
3006:
2981:
2966:
2947:
2891:
2882:
2876:
2503:{\displaystyle i=1,2,\dots ,n}
2411:
2405:
2384:
2378:
2331:
2316:
2272:
2243:
2219:
2190:
2142:
2136:
2091:
2085:
1735:
1679:
1457:
1442:
1333:
1277:
968:
892:
886:
861:
849:
843:
837:
568:
562:
541:
535:
532:
523:
516:
510:
501:
495:
482:
476:
473:
461:
385:
373:
335:
13:
1:
17508:Uniform boundedness principle
16769:. Cambridge University Press.
16557:Graduate Texts in Mathematics
16531:American Mathematical Society
16086:Narici & Beckenstein 2011
16069:
15605:Duality (projective geometry)
14121:Dual system § Transposes
14050:is naturally identified with
13770:is naturally identified with
12691: there exists some
12517: there exists some
11943:. This gives the topology on
11804:
11049:{\displaystyle V\oplus V^{*}}
9517:such that the restriction to
9189:with respect to two bases of
7992:determines an isomorphism of
7514:. But there is in general no
6899:of infinitely many copies of
6751:of infinitely many copies of
3300:, let its basis be chosen as
1658:be defined as the following:
115:continuous linear functionals
69:for short) consisting of all
17019:Eigenvalues and eigenvectors
16789:; Wolff, Manfred P. (1999).
16594:An Introduction to Manifolds
16209:Mac Lane & Birkhoff 1999
15776:{\displaystyle F^{*}\omega }
15697:{\displaystyle (\cdot )^{*}}
15664:An Introduction to Manifolds
15465:, or in other words, that Ψ(
14086:Riesz representation theorem
12186:{\displaystyle \varphi _{i}}
11430:{\displaystyle \mathbb {R} }
11023:. Thus while the direct sum
10038:belonging to some index set
9998:is any family of subsets of
8155:
8093:
8053:
7506:is finite-dimensional, then
6939:, and so the identification
6020:{\displaystyle \alpha \in A}
5383:{\displaystyle \mathbb {N} }
5325:{\displaystyle \alpha \in A}
5173:are parallel hyperplanes in
4950:then, by dimension reasons,
2736:{\displaystyle \lambda _{i}}
1022:is finite-dimensional, then
867:{\displaystyle \varphi (x)=}
733:The pairing of a functional
7:
15578:
15005:is separable, but its dual
13899:denotes the sequence whose
13593:
11226:This arises in physics via
8797:. The resulting functional
7124:is an infinite-dimensional
6740:{\displaystyle (F^{A})_{0}}
6677:{\displaystyle f_{\alpha }}
5920:{\displaystyle (F^{A})_{0}}
5364:, whose elements are those
5253:indexed by an infinite set
4314:{\displaystyle {\hat {E}}=}
2799:is linearly independent on
2687:(The functional applied in
1179:. This dual basis is a set
10:
18092:
17651:Invariant subspace problem
16284:{\displaystyle V^{T^{-1}}}
16254:Similarly, one can define
14114:
12882:is Hausdorff and the sets
12625:are contained in some set
11839:
10386:is finite-dimensional and
9387:{\displaystyle f\in V^{*}}
9141:of algebras, meaning that
9119:is finite-dimensional. If
9111:; this homomorphism is an
8658:
7488:Cantor's diagonal argument
7413:it suffices to prove that
7090:of larger dimension (as a
5146:of a linear functional in
3216:. Hence, it is a basis of
2855:{\displaystyle g\in V^{*}}
1413:{\displaystyle c^{i}\in F}
1255:, defined by the relation
1049:has the same dimension as
995:
263:(alternatively denoted by
18032:
18011:
18003:Transpose of a linear map
17995:
17974:
17895:
17844:
17781:
17740:
17664:
17643:
17602:
17541:
17483:
17429:
17371:
17364:
17297:
17259:
17215:
17152:
17104:
17046:
17035:
16931:
16913:
16791:Topological Vector Spaces
16779:Topological vector spaces
16767:Topological vector spaces
16674:Topological Vector Spaces
16441:Linear Algebra Done Right
15654:{\displaystyle V^{\lor }}
15355:, this map is in fact an
14440:onto its continuous dual
14218:The resulting functional
14117:Transpose of a linear map
11507:topological vector spaces
11503:discontinuous linear maps
11379:topological vector spaces
10585:first isomorphism theorem
8661:Transpose of a linear map
8655:Transpose of a linear map
7079:relating direct sums (of
6359:it takes on the basis of
5197:Infinite-dimensional case
4989:{\displaystyle 1\times n}
4914:{\displaystyle 1\times 1}
4868:{\displaystyle n\times 1}
2622:successively, lead us to
1529:bi-orthogonality property
1235:of linear functionals on
391:{\displaystyle \hom(V,F)}
283:{\displaystyle V^{\lor }}
17620:Spectrum of a C*-algebra
15906:for the algebraic dual (
15621:
15398:topological vector space
15351:As a consequence of the
15223:{\displaystyle \varphi }
14955:and it follows from the
14458:operators are linked by
14454:, the transpose and the
13936:{\displaystyle \varphi }
12206:{\displaystyle \varphi }
12090:{\displaystyle \varphi }
11818:topological vector space
11514:topological vector space
11219:. This is formalized by
10612:{\displaystyle f:V\to F}
10439:{\displaystyle W^{00}=W}
9568:{\displaystyle f|_{S}=0}
9510:{\displaystyle f:V\to F}
8884:{\displaystyle \varphi }
7758:If the bilinear form is
7533:into its dual space via
6027:, where such a function
5952:{\displaystyle f:A\to F}
1907:{\displaystyle x,y\in V}
747:{\displaystyle \varphi }
111:topological vector space
17717:Noncommutative geometry
15919:In many areas, such as
15796:{\displaystyle \omega }
14995:, then so is the space
14808:can be identified with
11670:the space of arbitrary
11277:unit of time (units of
11240:time–frequency analysis
11083:-dimensional space (if
10966:by the natural pairing
7492:Erdős–Kaplansky theorem
7075:is a special case of a
6697:{\displaystyle \alpha }
5590:{\displaystyle (x_{n})}
5557:{\displaystyle (a_{n})}
5043:in the plane, then the
992:Finite-dimensional case
17773:Tomita–Takesaki theory
17748:Approximation property
17692:Calculus of variations
17004:Row and column vectors
16312:
16285:
16026:
16006:
15986:
15900:
15873:
15847:
15826:
15797:
15777:
15746:
15725:
15698:
15661:used in this way, see
15655:
15550:
15434:to the algebraic dual
15342:
15232:
15224:
15202:
15025:natural transformation
14946:
14895:is the annihilator of
14873:, with range equal to
14849:; then, the transpose
14788:
14522:
14328:
14209:
14011:
13937:
13892:
13862:
13812:
13741:
13684:
13575:
13539:
13503:
13476:leads to a variant of
13470:
13441:
13421:
13397:
13377:
13345:
13325:
13301:
13277:
13242:
13161:
13124:
13100:
13080:
13056:
13032:
12993:
12876:
12846:
12762:
12735:
12649:
12619:
12589:
12555:
12485:
12455:
12435:
12411:
12384:
12232:
12207:
12193:tends to a functional
12187:
12158:
12131:
12111:
12091:
12068:
11984:
11957:
11937:
11913:
11889:
11869:
11836:Topologies on the dual
11795:tempered distributions
11787:
11751:
11724:
11692:
11664:
11632:
11602:
11566:
11541:topological dual space
11529:
11483:
11458:
11431:
11409:
11359:
11301:
11261:
11211:
11187:
11154:
11123:
11099:
11075:
11050:
11013:
10960:
10927:
10926:{\displaystyle v\in V}
10893:
10866:
10839:
10812:
10792:
10768:
10745:
10685:
10661:
10641:
10613:
10577:
10546:
10526:
10499:
10463:
10440:
10400:
10380:
10357:
10283:
10215:
10195:
10175:
10152:
10052:
10032:
10012:
9992:
9943:
9872:
9852:
9832:
9809:
9738:
9694:
9674:
9622:
9569:
9531:
9511:
9479:
9452:
9451:{\displaystyle s\in S}
9426:
9388:
9355:
9328:
9301:
9275:
9255:
9201:is represented by the
9182:is represented by the
9057:
8973:
8972:{\displaystyle v\in V}
8945:
8907:
8885:
8857:
8828:
8790:
8753:
8641:
8623:; and it is always an
8613:
8593:
8561:is defined as the map
8555:
8513:
8478:
8433:. In other words, if
8427:
8382:
8323:
8233:
8213:
8179:
8102:
8065:
7952:
7844:
7788:
7749:
7658:
7568:
7480:
7407:
7330:
7138:
7118:
7083:) to direct products.
7066:
6933:
6913:
6889:
6859:
6785:
6765:
6741:
6698:
6678:
6648:
6487:
6467:
6447:
6405:
6373:
6353:
6302:
6282:
6268:: a linear functional
6262:
6242:
6218:
6191:
6168:
6167:{\displaystyle v\in V}
6142:
6122:
6099:
6041:
6021:
5995:
5953:
5921:
5881:
5861:
5814:
5794:
5771:
5736:
5703:
5673:
5626:is sent to the number
5620:
5591:
5558:
5521:
5490:
5461:
5441:
5412:
5384:
5358:
5326:
5300:
5267:
5247:
5215:
5187:
5167:
5136:
5115:
5088:
5068:
5033:
5016:must be a row vector.
5010:
4990:
4964:
4944:
4915:
4889:
4869:
4843:
4823:
4803:
4783:
4763:
4739:
4710:
4683:
4663:
4625:
4590:
4460:
4432:
4408:
4378:
4315:
4240:
4174:
4145:
4122:
4063:
4010:
3981:
3952:
3860:
3670:
3617:
3564:
3511:
3454:
3425:
3396:
3294:
3265:
3237:
3210:
3183:
3124:
2856:
2820:
2793:
2737:
2710:
2681:
2616:
2596:
2504:
2460:
2418:
2291:
2162:
2111:
2060:
1984:
1908:
1876:
1810:
1780:
1652:
1596:
1517:
1482:
1414:
1378:
1249:
1229:
1169:
1142:
1122:
1063:
1043:
1016:
978:
917:
868:
819:
797:
776:
748:
710:
680:
679:{\displaystyle a\in F}
653:
652:{\displaystyle x\in V}
626:
584:
440:
419:
392:
345:
309:
284:
257:
235:(algebraic) dual space
227:
204:
91:
54:
17768:Banach–Mazur distance
17731:Generalized functions
17009:Row and column spaces
16954:Scalar multiplication
16313:
16311:{\displaystyle V^{T}}
16291:as the dual space to
16286:
16027:
16007:
15987:
15901:
15899:{\displaystyle V^{*}}
15874:
15848:
15827:
15798:
15778:
15747:
15726:
15724:{\displaystyle F^{*}}
15699:
15656:
15600:Duality (mathematics)
15551:
15343:
15225:
15203:
15022:
14947:
14830:denote the canonical
14789:
14543:, then the transpose
14523:
14420:Arzelà–Ascoli theorem
14412:, then the transpose
14329:
14210:
14012:
13938:
13893:
13863:
13813:
13742:
13664:
13576:
13540:
13504:
13471:
13442:
13422:
13398:
13378:
13346:
13326:
13302:
13278:
13243:
13162:
13125:
13101:
13081:
13057:
13033:
13002:form its local base.
12994:
12877:
12847:
12817: such that
12763:
12736:
12710: such that
12650:
12620:
12590:
12556:
12536: such that
12486:
12456:
12436:
12412:
12385:
12233:
12208:
12188:
12159:
12132:
12112:
12092:
12069:
11985:
11958:
11938:
11914:
11890:
11870:
11799:generalized functions
11788:
11752:
11725:
11693:
11665:
11633:
11603:
11567:
11537:continuous dual space
11530:
11484:
11459:
11457:{\displaystyle V^{*}}
11432:
11410:
11373:Continuous dual space
11360:
11302:
11262:
11212:
11188:
11155:
11153:{\displaystyle V^{*}}
11124:
11100:
11076:
11051:
11014:
10961:
10928:
10894:
10892:{\displaystyle B^{0}}
10867:
10865:{\displaystyle A^{0}}
10840:
10838:{\displaystyle V^{*}}
10813:
10793:
10769:
10746:
10686:
10662:
10642:
10614:
10578:
10547:
10527:
10500:
10464:
10441:
10401:
10381:
10358:
10284:
10216:
10196:
10176:
10153:
10053:
10033:
10013:
9993:
9944:
9873:
9853:
9833:
9810:
9739:
9695:
9675:
9623:
9577:orthogonal complement
9570:
9532:
9512:
9480:
9478:{\displaystyle S^{0}}
9453:
9427:
9389:
9356:
9354:{\displaystyle S^{0}}
9329:
9327:{\displaystyle V^{*}}
9302:
9276:
9256:
9161:contravariant functor
9155:. In the language of
9058:
8974:
8946:
8908:
8886:
8858:
8856:{\displaystyle V^{*}}
8829:
8791:
8754:
8642:
8614:
8612:{\displaystyle \Psi }
8594:
8556:
8514:
8479:
8428:
8383:
8324:
8239:into the double dual
8234:
8214:
8212:{\displaystyle \Psi }
8180:
8103:
8066:
7953:
7845:
7789:
7787:{\displaystyle \Phi }
7750:
7659:
7569:
7481:
7408:
7331:
7139:
7119:
7067:
6934:
6914:
6890:
6888:{\displaystyle F^{A}}
6860:
6786:
6766:
6742:
6699:
6679:
6649:
6488:
6468:
6448:
6406:
6374:
6354:
6303:
6283:
6263:
6243:
6219:
6217:{\displaystyle F^{A}}
6192:
6169:
6143:
6123:
6100:
6042:
6022:
5996:
5954:
5922:
5882:
5862:
5815:
5795:
5772:
5737:
5704:
5702:{\displaystyle x_{n}}
5674:
5621:
5592:
5559:
5522:
5491:
5462:
5442:
5413:
5385:
5359:
5327:
5301:
5268:
5248:
5216:
5188:
5168:
5166:{\displaystyle V^{*}}
5137:
5116:
5114:{\displaystyle V^{*}}
5089:
5069:
5067:{\displaystyle V^{*}}
5034:
5011:
4991:
4965:
4945:
4916:
4890:
4870:
4844:
4824:
4804:
4784:
4764:
4747:matrix multiplication
4740:
4711:
4684:
4664:
4626:
4591:
4466:to be represented as
4461:
4433:
4409:
4407:{\displaystyle I_{n}}
4379:
4316:
4241:
4175:
4146:
4123:
4064:
4011:
3982:
3953:
3861:
3671:
3618:
3565:
3512:
3455:
3426:
3397:
3295:
3266:
3238:
3236:{\displaystyle V^{*}}
3211:
3209:{\displaystyle V^{*}}
3184:
3125:
2857:
2821:
2819:{\displaystyle V^{*}}
2794:
2738:
2711:
2682:
2617:
2597:
2505:
2461:
2419:
2292:
2163:
2112:
2061:
1985:
1909:
1877:
1811:
1809:{\displaystyle V^{*}}
1789:These are a basis of
1781:
1653:
1597:
1518:
1483:
1415:
1379:
1250:
1230:
1170:
1168:{\displaystyle V^{*}}
1143:
1123:
1064:
1044:
1042:{\displaystyle V^{*}}
1017:
979:
918:
869:
820:
798:
777:
775:{\displaystyle V^{*}}
749:
716:are sometimes called
711:
709:{\displaystyle V^{*}}
681:
654:
627:
585:
441:
420:
418:{\displaystyle V^{*}}
393:
346:
310:
285:
258:
256:{\displaystyle V^{*}}
228:
205:
119:continuous dual space
109:. When defined for a
92:
55:
17513:Kakutani fixed-point
17498:Riesz representation
17144:Gram–Schmidt process
17096:Gaussian elimination
16497:Halmos, Paul Richard
16295:
16258:
16016:
15996:
15965:
15883:
15858:
15837:
15811:
15787:
15757:
15736:
15708:
15675:
15638:
15498:
15263:
15246:from a normed space
15214:
15084:
14906:
14727:
14465:
14360:is equal to that of
14276:
14152:
14064:sequences, with the
13957:
13927:
13873:
13828:
13778:
13634:
13581:is endowed with the
13560:
13524:
13509:is endowed with the
13488:
13478:reflexivity property
13455:
13431:
13407:
13387:
13362:
13335:
13311:
13291:
13285:totally bounded sets
13262:
13176:
13146:
13114:
13090:
13066:
13046:
13017:
12889:
12861:
12774:
12748:
12661:
12629:
12599:
12569:
12497:
12465:
12461:belongs to some set
12445:
12425:
12397:
12245:
12217:
12197:
12170:
12141:
12137:runs over the class
12121:
12101:
12081:
12001:
11967:
11947:
11927:
11899:
11879:
11854:
11765:
11734:
11702:
11678:
11642:
11618:
11576:
11551:
11519:
11468:
11441:
11419:
11389:
11321:
11283:
11251:
11228:dimensional analysis
11201:
11186:{\displaystyle (-n)}
11168:
11137:
11113:
11089:
11062:
11027:
10970:
10937:
10911:
10903:Dimensional analysis
10876:
10849:
10822:
10802:
10782:
10758:
10698:
10675:
10651:
10623:
10591:
10559:
10536:
10516:
10473:
10453:
10417:
10390:
10370:
10299:
10228:
10205:
10185:
10165:
10065:
10042:
10022:
10002:
9956:
9885:
9862:
9842:
9822:
9751:
9704:
9684:
9632:
9586:
9541:
9521:
9489:
9462:
9436:
9398:
9365:
9338:
9311:
9291:
9265:
9245:
8987:
8957:
8922:
8897:
8875:
8840:
8802:
8767:
8714:
8631:
8603:
8565:
8523:
8488:
8437:
8392:
8333:
8243:
8223:
8203:
8129:
8078:
8007:
7976:If the vector space
7862:
7815:
7778:
7674:
7610:
7540:
7417:
7347:
7155:
7128:
7108:
6946:
6923:
6903:
6872:
6798:
6775:
6755:
6711:
6688:
6661:
6500:
6477:
6457:
6415:
6383:
6363:
6312:
6292:
6272:
6252:
6232:
6201:
6181:
6152:
6132:
6112:
6054:
6031:
6005:
5963:
5931:
5891:
5871:
5824:
5820:: a choice of basis
5804:
5784:
5750:
5717:
5686:
5633:
5601:
5568:
5535:
5500:
5471:
5451:
5422:
5394:
5372:
5339:
5310:
5281:
5275:linearly independent
5257:
5228:
5205:
5177:
5150:
5126:
5098:
5078:
5051:
5023:
5000:
4974:
4954:
4943:{\displaystyle Mx=y}
4925:
4899:
4879:
4853:
4833:
4813:
4793:
4773:
4753:
4720:
4700:
4673:
4644:
4603:
4473:
4442:
4422:
4391:
4328:
4250:
4184:
4155:
4135:
4073:
4020:
3991:
3962:
3873:
3683:
3627:
3574:
3521:
3468:
3435:
3406:
3304:
3275:
3255:
3220:
3193:
3137:
2870:
2833:
2803:
2747:
2720:
2691:
2626:
2606:
2517:
2470:
2428:
2301:
2172:
2121:
2070:
1994:
1918:
1886:
1824:
1793:
1664:
1606:
1602:the basis of V. Let
1550:
1495:
1427:
1391:
1262:
1239:
1183:
1152:
1132:
1076:
1053:
1026:
1006:
931:
880:
831:
809:
787:
759:
738:
693:
664:
637:
597:
454:
430:
402:
364:
323:
294:
267:
240:
217:
194:
183:Algebraic dual space
107:algebraic dual space
78:
61:has a corresponding
44:
18061:Functional analysis
18040:Biorthogonal system
17872:Operator topologies
17697:Functional calculus
17656:Mahler's conjecture
17635:Von Neumann algebra
17349:Functional analysis
17274:Numerical stability
17154:Multilinear algebra
17129:Inner product space
16979:Linear independence
16862:"Dual Vector Space"
16787:Schaefer, Helmut H.
16775:Schaefer, Helmut H.
16735:Functional Analysis
16706:Functional Analysis
16523:Katznelson, Yitzhak
16469:. Springer-Verlag.
16088:, pp. 225–273.
15925:⟨·,·⟩
15807:, p. 20) uses
15469:) is continuous on
15353:Hahn–Banach theorem
14957:Hahn–Banach theorem
14634:is continuous from
14549:is continuous when
14381:Hahn–Banach theorem
14060:(consisting of all
14039:Hölder's inequality
13774:: given an element
13256:stereotype topology
13132:normed vector space
12798: and all
12779: for all
12666: for all
12502: for all
12250: for all
11895:. Fix a collection
11300:{\displaystyle 1/t}
11021:reducing a fraction
10845:is a direct sum of
10640:{\displaystyle V/W}
10576:{\displaystyle V/W}
10144:
9858:are two subsets of
9135:composition of maps
8649:natural isomorphism
7990:⟨·,·⟩
7529:gives a mapping of
7523:⟨·,·⟩
7516:natural isomorphism
7343:. For proving that
6868:On the other hand,
6379:, and any function
5496:is (isomorphic to)
4789:into a real number
1512:
1477:
169:transponierter Raum
146:functional analysis
18076:Linear functionals
17722:Riemann hypothesis
17421:Topological vector
16984:Linear combination
16859:Weisstein, Eric W.
16643:Misner, Charles W.
16616:Mac Lane, Saunders
16491:. Springer-Verlag.
16437:Axler, Sheldon Jay
16308:
16281:
16022:
16002:
15982:
15927:is reserved for a
15896:
15872:{\displaystyle V'}
15869:
15843:
15825:{\displaystyle V'}
15822:
15793:
15773:
15742:
15721:
15694:
15651:
15615:Reciprocal lattice
15610:Pontryagin duality
15546:
15338:
15233:
15231:in the dual space.
15220:
15198:
14983:Further properties
14942:
14838:onto the quotient
14812:, and the dual of
14784:
14518:
14324:
14205:
14007:
13986:
13933:
13888:
13858:
13808:
13750:Define the number
13737:
13574:{\displaystyle V'}
13571:
13538:{\displaystyle V'}
13535:
13502:{\displaystyle V'}
13499:
13469:{\displaystyle V'}
13466:
13437:
13417:
13393:
13376:{\displaystyle V'}
13373:
13341:
13321:
13297:
13276:{\displaystyle V'}
13273:
13238:
13212:
13160:{\displaystyle V'}
13157:
13120:
13096:
13076:
13052:
13031:{\displaystyle V'}
13028:
12989:
12875:{\displaystyle V'}
12872:
12842:
12758:
12731:
12645:
12615:
12585:
12551:
12481:
12451:
12431:
12407:
12380:
12375:
12314:
12231:{\displaystyle V'}
12228:
12203:
12183:
12154:
12127:
12107:
12087:
12064:
12038:
11980:
11953:
11933:
11909:
11885:
11868:{\displaystyle V'}
11865:
11783:
11747:
11720:
11688:
11660:
11628:
11598:
11565:{\displaystyle V'}
11562:
11525:
11491:finite-dimensional
11482:{\displaystyle V'}
11479:
11454:
11427:
11405:
11377:When dealing with
11355:
11297:
11257:
11238:, or more broadly
11221:tensor contraction
11207:
11183:
11150:
11119:
11095:
11074:{\displaystyle 2n}
11071:
11046:
11009:
10956:
10923:
10889:
10862:
10835:
10808:
10788:
10764:
10741:
10681:
10657:
10637:
10609:
10573:
10542:
10522:
10495:
10459:
10449:after identifying
10436:
10396:
10376:
10353:
10279:
10211:
10191:
10171:
10148:
10130:
10129:
10089:
10048:
10028:
10008:
9988:
9939:
9868:
9848:
9828:
9805:
9734:
9690:
9670:
9618:
9565:
9527:
9507:
9475:
9448:
9425:{\displaystyle =0}
9422:
9384:
9351:
9324:
9297:
9271:
9251:
9221:is represented by
9178:If the linear map
9056:{\displaystyle =,}
9053:
8969:
8941:
8903:
8881:
8853:
8824:
8786:
8749:
8637:
8609:
8589:
8551:
8509:
8474:
8423:
8378:
8319:
8229:
8209:
8175:
8098:
8061:
8000:of the dual space
7986:sesquilinear forms
7948:
7840:
7784:
7745:
7654:
7564:
7476:
7403:
7326:
7134:
7114:
7062:
7045:
7016:
6983:
6929:
6909:
6885:
6855:
6848:
6781:
6761:
6737:
6694:
6674:
6644:
6620:
6572:
6526:
6483:
6463:
6443:
6401:
6369:
6349:
6298:
6278:
6258:
6238:
6214:
6187:
6177:The dual space of
6164:
6138:
6118:
6095:
6072:
6037:
6017:
5991:
5949:
5917:
5877:
5857:
5810:
5790:
5767:
5744:countably infinite
5732:
5699:
5669:
5645:
5616:
5587:
5554:
5517:
5486:
5457:
5437:
5408:
5380:
5354:
5322:
5296:
5263:
5243:
5211:
5183:
5163:
5132:
5111:
5084:
5064:
5029:
5006:
4986:
4960:
4940:
4911:
4885:
4865:
4839:
4819:
4799:
4779:
4759:
4735:
4706:
4679:
4659:
4621:
4586:
4544:
4493:
4456:
4428:
4404:
4374:
4311:
4239:{\displaystyle E=}
4236:
4170:
4141:
4118:
4059:
4006:
3977:
3948:
3856:
3847:
3808:
3744:
3666:
3613:
3560:
3507:
3450:
3421:
3392:
3290:
3261:
3233:
3206:
3179:
3120:
2852:
2816:
2789:
2733:
2706:
2677:
2612:
2592:
2500:
2456:
2414:
2287:
2158:
2107:
2056:
1980:
1904:
1872:
1806:
1776:
1648:
1592:
1513:
1498:
1478:
1463:
1410:
1374:
1245:
1225:
1165:
1138:
1118:
1059:
1039:
1012:
974:
913:
864:
815:
793:
772:
755:in the dual space
744:
706:
676:
649:
622:
580:
578:
436:
415:
388:
354:linear functionals
341:
308:{\displaystyle V'}
305:
280:
253:
223:
200:
130:finite-dimensional
90:{\displaystyle V,}
87:
50:
18048:
18047:
17937:in Hilbert spaces
17799:
17798:
17702:Integral operator
17479:
17478:
17315:
17314:
17182:Geometric algebra
17139:Kronecker product
16974:Linear projection
16959:Vector projection
16834:978-0-486-45352-1
16804:978-1-4612-7155-0
16749:978-0-07-054236-5
16657:. W. H. Freeman.
16620:Birkhoff, Garrett
16607:978-1-4419-7400-6
16566:978-0-387-95385-4
16540:978-0-8218-4419-9
16485:Bourbaki, Nicolas
16463:Bourbaki, Nicolas
16454:978-3-319-11079-0
16025:{\displaystyle V}
16005:{\displaystyle V}
15929:sesquilinear form
15921:quantum mechanics
15846:{\displaystyle V}
15745:{\displaystyle F}
15320:
14235:. The assignment
14098:quantum mechanics
13977:
13725:
13547:stereotype spaces
13440:{\displaystyle V}
13396:{\displaystyle V}
13344:{\displaystyle V}
13300:{\displaystyle V}
13191:
13123:{\displaystyle V}
13099:{\displaystyle V}
13055:{\displaystyle V}
12974:
12938:
12932:
12910:
12904:
12818:
12799:
12780:
12711:
12692:
12667:
12537:
12518:
12503:
12454:{\displaystyle V}
12434:{\displaystyle x}
12360:
12299:
12251:
12130:{\displaystyle A}
12110:{\displaystyle V}
12023:
11956:{\displaystyle V}
11936:{\displaystyle V}
11888:{\displaystyle V}
11528:{\displaystyle V}
11260:{\displaystyle t}
11210:{\displaystyle V}
11122:{\displaystyle n}
11098:{\displaystyle V}
10811:{\displaystyle B}
10791:{\displaystyle A}
10778:of two subspaces
10767:{\displaystyle V}
10684:{\displaystyle f}
10660:{\displaystyle W}
10545:{\displaystyle V}
10532:is a subspace of
10525:{\displaystyle W}
10507:Galois connection
10462:{\displaystyle W}
10399:{\displaystyle W}
10379:{\displaystyle V}
10214:{\displaystyle V}
10201:are subspaces of
10194:{\displaystyle B}
10174:{\displaystyle A}
10161:In particular if
10114:
10074:
10051:{\displaystyle I}
10031:{\displaystyle i}
10011:{\displaystyle V}
9871:{\displaystyle V}
9851:{\displaystyle B}
9831:{\displaystyle A}
9693:{\displaystyle V}
9530:{\displaystyle S}
9300:{\displaystyle S}
9274:{\displaystyle V}
9254:{\displaystyle S}
8906:{\displaystyle f}
8640:{\displaystyle V}
8295:
8232:{\displaystyle V}
8158:
8096:
8056:
7998:complex conjugate
7798:to a subspace of
7510:is isomorphic to
7137:{\displaystyle F}
7117:{\displaystyle V}
7030:
7001:
6968:
6932:{\displaystyle A}
6912:{\displaystyle F}
6833:
6784:{\displaystyle A}
6764:{\displaystyle F}
6605:
6557:
6511:
6486:{\displaystyle V}
6466:{\displaystyle T}
6372:{\displaystyle V}
6301:{\displaystyle V}
6281:{\displaystyle T}
6261:{\displaystyle F}
6241:{\displaystyle A}
6190:{\displaystyle V}
6141:{\displaystyle f}
6121:{\displaystyle V}
6057:
6040:{\displaystyle f}
5880:{\displaystyle V}
5813:{\displaystyle F}
5793:{\displaystyle V}
5636:
5460:{\displaystyle i}
5266:{\displaystyle A}
5214:{\displaystyle V}
5186:{\displaystyle V}
5135:{\displaystyle V}
5087:{\displaystyle V}
5047:of an element of
5032:{\displaystyle V}
5009:{\displaystyle M}
4996:matrix; that is,
4963:{\displaystyle M}
4888:{\displaystyle y}
4842:{\displaystyle x}
4822:{\displaystyle M}
4802:{\displaystyle y}
4782:{\displaystyle x}
4762:{\displaystyle n}
4709:{\displaystyle n}
4682:{\displaystyle n}
4535:
4484:
4431:{\displaystyle n}
4348:
4341:
4262:
4144:{\displaystyle V}
4131:In general, when
3264:{\displaystyle V}
3248:
3247:
2829:Lastly, consider
2615:{\displaystyle V}
1248:{\displaystyle V}
1141:{\displaystyle V}
1062:{\displaystyle V}
1015:{\displaystyle V}
818:{\displaystyle V}
796:{\displaystyle x}
439:{\displaystyle F}
398:. The dual space
226:{\displaystyle F}
203:{\displaystyle V}
63:dual vector space
53:{\displaystyle V}
18:Dual vector space
16:(Redirected from
18083:
18071:Duality theories
18019:Saturated family
17917:Ultraweak/Weak-*
17826:
17819:
17812:
17803:
17802:
17789:
17788:
17707:Jones polynomial
17625:Operator algebra
17369:
17368:
17342:
17335:
17328:
17319:
17318:
17305:
17304:
17187:Exterior algebra
17124:Hadamard product
17041:
17029:Linear equations
16900:
16893:
16886:
16877:
16876:
16872:
16871:
16846:
16821:Trèves, François
16816:
16782:
16770:
16761:
16725:
16709:
16695:
16668:
16651:Wheeler, John A.
16637:
16611:
16596:(2nd ed.).
16585:
16544:
16518:
16503:(2nd ed.).
16492:
16480:
16458:
16443:(3rd ed.).
16423:
16417:
16411:
16405:
16399:
16393:
16387:
16381:
16375:
16369:
16363:
16357:
16351:
16345:
16336:
16330:
16321:
16320:
16317:
16315:
16314:
16309:
16307:
16306:
16290:
16288:
16287:
16282:
16280:
16279:
16278:
16277:
16242:
16236:
16230:
16224:
16218:
16212:
16206:
16200:
16199:
16181:
16175:
16169:
16163:
16158:
16152:
16146:
16140:
16134:
16128:
16122:
16116:
16110:
16101:
16095:
16089:
16083:
16063:
16052:
16046:
16039:
16033:
16031:
16029:
16028:
16023:
16011:
16009:
16008:
16003:
15991:
15989:
15988:
15983:
15981:
15980:
15979:
15973:
15955:
15942:
15940:
15926:
15917:
15911:
15905:
15903:
15902:
15897:
15895:
15894:
15878:
15876:
15875:
15870:
15868:
15852:
15850:
15849:
15844:
15831:
15829:
15828:
15823:
15821:
15802:
15800:
15799:
15794:
15782:
15780:
15779:
15774:
15769:
15768:
15751:
15749:
15748:
15743:
15730:
15728:
15727:
15722:
15720:
15719:
15703:
15701:
15700:
15695:
15693:
15692:
15660:
15658:
15657:
15652:
15650:
15649:
15632:
15570:
15564:
15555:
15553:
15552:
15547:
15514:
15490:
15484:
15474:
15464:
15454:
15448:
15439:
15413:
15380:
15370:
15347:
15345:
15344:
15339:
15334:
15318:
15255:
15245:
15229:
15227:
15226:
15221:
15207:
15205:
15204:
15199:
15179:
15178:
15166:
15165:
15132:
15131:
15116:
15105:
15104:
15078:
15044:
15010:
15004:
14998:
14990:
14978:
14964:
14951:
14949:
14948:
14943:
14941:
14940:
14925:
14894:
14872:
14866:
14854:
14848:
14825:
14807:
14793:
14791:
14790:
14785:
14759:
14739:
14738:
14719:
14693:
14678:
14663:
14648:
14633:
14628:. The transpose
14623:
14604:
14586:
14580:
14570:
14560:
14554:
14548:
14527:
14525:
14524:
14519:
14514:
14513:
14501:
14490:
14489:
14477:
14476:
14445:
14417:
14392:
14378:
14359:
14333:
14331:
14330:
14325:
14320:
14309:
14298:
14260:
14254:
14240:
14234:
14228:
14214:
14212:
14211:
14206:
14201:
14162:
14144:
14134:
14094:bra–ket notation
14080:
14055:
14049:
14036:
14016:
14014:
14013:
14008:
14006:
14005:
13996:
13995:
13985:
13970:
13949:
13942:
13940:
13939:
13934:
13921:
13902:
13897:
13895:
13894:
13889:
13887:
13886:
13881:
13867:
13865:
13864:
13859:
13851:
13850:
13845:
13824:is the sequence
13823:
13817:
13815:
13814:
13809:
13807:
13799:
13798:
13765:
13746:
13744:
13743:
13738:
13727:
13726:
13718:
13716:
13712:
13711:
13710:
13705:
13699:
13698:
13689:
13683:
13678:
13654:
13653:
13644:
13626:
13580:
13578:
13577:
13572:
13570:
13544:
13542:
13541:
13536:
13534:
13508:
13506:
13505:
13500:
13498:
13475:
13473:
13472:
13467:
13465:
13446:
13444:
13443:
13438:
13426:
13424:
13423:
13418:
13416:
13415:
13402:
13400:
13399:
13394:
13382:
13380:
13379:
13374:
13372:
13350:
13348:
13347:
13342:
13330:
13328:
13327:
13322:
13320:
13319:
13306:
13304:
13303:
13298:
13282:
13280:
13279:
13274:
13272:
13247:
13245:
13244:
13239:
13234:
13217:
13211:
13166:
13164:
13163:
13158:
13156:
13134:(for example, a
13129:
13127:
13126:
13121:
13105:
13103:
13102:
13097:
13085:
13083:
13082:
13077:
13075:
13074:
13061:
13059:
13058:
13053:
13037:
13035:
13034:
13029:
13027:
12998:
12996:
12995:
12990:
12988:
12987:
12975:
12972:
12966:
12962:
12955:
12954:
12936:
12930:
12929:
12908:
12902:
12901:
12900:
12881:
12879:
12878:
12873:
12871:
12851:
12849:
12848:
12843:
12838:
12837:
12819:
12816:
12813:
12812:
12800:
12797:
12794:
12793:
12781:
12778:
12767:
12765:
12764:
12759:
12757:
12756:
12740:
12738:
12737:
12732:
12712:
12709:
12706:
12705:
12693:
12690:
12687:
12686:
12668:
12665:
12654:
12652:
12651:
12646:
12644:
12643:
12624:
12622:
12621:
12616:
12614:
12613:
12594:
12592:
12591:
12586:
12584:
12583:
12560:
12558:
12557:
12552:
12538:
12535:
12532:
12531:
12519:
12516:
12504:
12501:
12490:
12488:
12487:
12482:
12480:
12479:
12460:
12458:
12457:
12452:
12440:
12438:
12437:
12432:
12416:
12414:
12413:
12408:
12406:
12405:
12389:
12387:
12386:
12381:
12376:
12374:
12358:
12329:
12328:
12319:
12313:
12295:
12294:
12279:
12278:
12265:
12264:
12252:
12249:
12237:
12235:
12234:
12229:
12227:
12212:
12210:
12209:
12204:
12192:
12190:
12189:
12184:
12182:
12181:
12163:
12161:
12160:
12155:
12150:
12149:
12136:
12134:
12133:
12128:
12116:
12114:
12113:
12108:
12096:
12094:
12093:
12088:
12073:
12071:
12070:
12065:
12060:
12043:
12037:
12019:
12018:
11989:
11987:
11986:
11981:
11976:
11975:
11962:
11960:
11959:
11954:
11942:
11940:
11939:
11934:
11918:
11916:
11915:
11910:
11908:
11907:
11894:
11892:
11891:
11886:
11874:
11872:
11871:
11866:
11864:
11831:
11823:
11812:
11792:
11790:
11789:
11784:
11779:
11775:
11774:
11756:
11754:
11753:
11748:
11743:
11742:
11729:
11727:
11726:
11721:
11716:
11712:
11711:
11697:
11695:
11694:
11689:
11687:
11686:
11669:
11667:
11666:
11661:
11656:
11652:
11651:
11637:
11635:
11634:
11629:
11627:
11626:
11607:
11605:
11604:
11599:
11597:
11596:
11571:
11569:
11568:
11563:
11561:
11534:
11532:
11531:
11526:
11488:
11486:
11485:
11480:
11478:
11463:
11461:
11460:
11455:
11453:
11452:
11436:
11434:
11433:
11428:
11426:
11414:
11412:
11411:
11406:
11404:
11396:
11364:
11362:
11361:
11356:
11348:
11347:
11308:
11306:
11304:
11303:
11298:
11293:
11268:
11266:
11264:
11263:
11258:
11236:Fourier analysis
11218:
11216:
11214:
11213:
11208:
11194:
11192:
11190:
11189:
11184:
11161:
11159:
11157:
11156:
11151:
11149:
11148:
11130:
11128:
11126:
11125:
11120:
11106:
11104:
11102:
11101:
11096:
11082:
11080:
11078:
11077:
11072:
11055:
11053:
11052:
11047:
11045:
11044:
11018:
11016:
11015:
11010:
10965:
10963:
10962:
10957:
10955:
10954:
10932:
10930:
10929:
10924:
10898:
10896:
10895:
10890:
10888:
10887:
10871:
10869:
10868:
10863:
10861:
10860:
10844:
10842:
10841:
10836:
10834:
10833:
10817:
10815:
10814:
10809:
10797:
10795:
10794:
10789:
10773:
10771:
10770:
10765:
10750:
10748:
10747:
10742:
10737:
10736:
10724:
10723:
10711:
10690:
10688:
10687:
10682:
10666:
10664:
10663:
10658:
10646:
10644:
10643:
10638:
10633:
10619:factors through
10618:
10616:
10615:
10610:
10582:
10580:
10579:
10574:
10569:
10551:
10549:
10548:
10543:
10531:
10529:
10528:
10523:
10504:
10502:
10501:
10496:
10494:
10493:
10468:
10466:
10465:
10460:
10445:
10443:
10442:
10437:
10429:
10428:
10405:
10403:
10402:
10397:
10385:
10383:
10382:
10377:
10362:
10360:
10359:
10354:
10349:
10348:
10336:
10335:
10323:
10322:
10288:
10286:
10285:
10280:
10278:
10277:
10265:
10264:
10252:
10251:
10220:
10218:
10217:
10212:
10200:
10198:
10197:
10192:
10180:
10178:
10177:
10172:
10157:
10155:
10154:
10149:
10143:
10138:
10128:
10110:
10109:
10104:
10100:
10099:
10098:
10088:
10057:
10055:
10054:
10049:
10037:
10035:
10034:
10029:
10017:
10015:
10014:
10009:
9997:
9995:
9994:
9989:
9987:
9986:
9971:
9970:
9948:
9946:
9945:
9940:
9935:
9934:
9910:
9909:
9897:
9896:
9877:
9875:
9874:
9869:
9857:
9855:
9854:
9849:
9837:
9835:
9834:
9829:
9814:
9812:
9811:
9806:
9801:
9800:
9788:
9787:
9775:
9774:
9743:
9741:
9740:
9735:
9699:
9697:
9696:
9691:
9679:
9677:
9676:
9671:
9669:
9668:
9644:
9643:
9627:
9625:
9624:
9619:
9617:
9616:
9604:
9603:
9574:
9572:
9571:
9566:
9558:
9557:
9552:
9536:
9534:
9533:
9528:
9516:
9514:
9513:
9508:
9484:
9482:
9481:
9476:
9474:
9473:
9457:
9455:
9454:
9449:
9431:
9429:
9428:
9423:
9393:
9391:
9390:
9385:
9383:
9382:
9360:
9358:
9357:
9352:
9350:
9349:
9333:
9331:
9330:
9325:
9323:
9322:
9306:
9304:
9303:
9298:
9280:
9278:
9277:
9272:
9260:
9258:
9257:
9252:
9154:
9139:antihomomorphism
9128:
9090:
9062:
9060:
9059:
9054:
9002:
9001:
8978:
8976:
8975:
8970:
8950:
8948:
8947:
8942:
8940:
8939:
8912:
8910:
8909:
8904:
8890:
8888:
8887:
8882:
8862:
8860:
8859:
8854:
8852:
8851:
8833:
8831:
8830:
8825:
8814:
8813:
8795:
8793:
8792:
8787:
8785:
8784:
8758:
8756:
8755:
8750:
8726:
8725:
8706:
8678:
8646:
8644:
8643:
8638:
8618:
8616:
8615:
8610:
8598:
8596:
8595:
8590:
8588:
8587:
8582:
8560:
8558:
8557:
8552:
8550:
8549:
8518:
8516:
8515:
8510:
8483:
8481:
8480:
8475:
8467:
8466:
8454:
8453:
8448:
8432:
8430:
8429:
8424:
8422:
8421:
8387:
8385:
8384:
8379:
8328:
8326:
8325:
8320:
8315:
8293:
8280:
8279:
8258:
8257:
8238:
8236:
8235:
8230:
8218:
8216:
8215:
8210:
8184:
8182:
8181:
8176:
8159:
8151:
8121:
8107:
8105:
8104:
8099:
8097:
8092:
8091:
8082:
8070:
8068:
8067:
8062:
8057:
8052:
8051:
8042:
8031:
8030:
7991:
7957:
7955:
7954:
7949:
7886:
7885:
7850:
7849:
7847:
7846:
7841:
7839:
7838:
7793:
7791:
7790:
7785:
7754:
7752:
7751:
7746:
7723:
7719:
7703:
7702:
7663:
7661:
7660:
7655:
7653:
7652:
7634:
7633:
7602:
7590:
7573:
7571:
7570:
7565:
7524:
7485:
7483:
7482:
7477:
7472:
7464:
7463:
7451:
7440:
7432:
7424:
7412:
7410:
7409:
7404:
7396:
7395:
7383:
7360:
7335:
7333:
7332:
7327:
7319:
7311:
7303:
7295:
7294:
7282:
7271:
7263:
7249:
7244:
7243:
7234:
7226:
7225:
7224:
7216:
7210:
7201:
7193:
7185:
7168:
7146:cardinal numbers
7143:
7141:
7140:
7135:
7123:
7121:
7120:
7115:
7071:
7069:
7068:
7063:
7061:
7060:
7044:
7026:
7025:
7015:
6997:
6996:
6991:
6987:
6982:
6958:
6957:
6938:
6936:
6935:
6930:
6918:
6916:
6915:
6910:
6894:
6892:
6891:
6886:
6884:
6883:
6864:
6862:
6861:
6856:
6847:
6829:
6828:
6819:
6818:
6790:
6788:
6787:
6782:
6770:
6768:
6767:
6762:
6746:
6744:
6743:
6738:
6736:
6735:
6726:
6725:
6703:
6701:
6700:
6695:
6683:
6681:
6680:
6675:
6673:
6672:
6653:
6651:
6650:
6645:
6640:
6639:
6630:
6629:
6619:
6598:
6597:
6582:
6581:
6571:
6553:
6549:
6548:
6547:
6542:
6536:
6535:
6525:
6492:
6490:
6489:
6484:
6472:
6470:
6469:
6464:
6452:
6450:
6449:
6444:
6442:
6441:
6410:
6408:
6407:
6402:
6378:
6376:
6375:
6370:
6358:
6356:
6355:
6350:
6345:
6344:
6339:
6324:
6323:
6307:
6305:
6304:
6299:
6287:
6285:
6284:
6279:
6267:
6265:
6264:
6259:
6247:
6245:
6244:
6239:
6223:
6221:
6220:
6215:
6213:
6212:
6196:
6194:
6193:
6188:
6173:
6171:
6170:
6165:
6147:
6145:
6144:
6139:
6127:
6125:
6124:
6119:
6104:
6102:
6101:
6096:
6094:
6093:
6088:
6082:
6081:
6071:
6046:
6044:
6043:
6038:
6026:
6024:
6023:
6018:
6000:
5998:
5997:
5992:
5975:
5974:
5958:
5956:
5955:
5950:
5926:
5924:
5923:
5918:
5916:
5915:
5906:
5905:
5886:
5884:
5883:
5878:
5866:
5864:
5863:
5858:
5841:
5840:
5835:
5819:
5817:
5816:
5811:
5799:
5797:
5796:
5791:
5776:
5774:
5773:
5768:
5766:
5765:
5764:
5758:
5741:
5739:
5738:
5733:
5731:
5730:
5725:
5708:
5706:
5705:
5700:
5698:
5697:
5678:
5676:
5675:
5670:
5665:
5664:
5655:
5654:
5644:
5625:
5623:
5622:
5617:
5615:
5614:
5609:
5596:
5594:
5593:
5588:
5583:
5582:
5563:
5561:
5560:
5555:
5550:
5549:
5526:
5524:
5523:
5518:
5516:
5515:
5514:
5508:
5495:
5493:
5492:
5487:
5485:
5484:
5479:
5466:
5464:
5463:
5458:
5446:
5444:
5443:
5438:
5436:
5435:
5430:
5417:
5415:
5414:
5409:
5407:
5389:
5387:
5386:
5381:
5379:
5363:
5361:
5360:
5355:
5353:
5352:
5347:
5331:
5329:
5328:
5323:
5305:
5303:
5302:
5297:
5295:
5294:
5289:
5272:
5270:
5269:
5264:
5252:
5250:
5249:
5244:
5242:
5241:
5236:
5220:
5218:
5217:
5212:
5192:
5190:
5189:
5184:
5172:
5170:
5169:
5164:
5162:
5161:
5141:
5139:
5138:
5133:
5120:
5118:
5117:
5112:
5110:
5109:
5093:
5091:
5090:
5085:
5073:
5071:
5070:
5065:
5063:
5062:
5038:
5036:
5035:
5030:
5015:
5013:
5012:
5007:
4995:
4993:
4992:
4987:
4969:
4967:
4966:
4961:
4949:
4947:
4946:
4941:
4920:
4918:
4917:
4912:
4894:
4892:
4891:
4886:
4874:
4872:
4871:
4866:
4848:
4846:
4845:
4840:
4828:
4826:
4825:
4820:
4808:
4806:
4805:
4800:
4788:
4786:
4785:
4780:
4768:
4766:
4765:
4760:
4744:
4742:
4741:
4736:
4734:
4733:
4728:
4715:
4713:
4712:
4707:
4688:
4686:
4685:
4680:
4668:
4666:
4665:
4660:
4658:
4657:
4652:
4630:
4628:
4627:
4622:
4595:
4593:
4592:
4587:
4582:
4581:
4576:
4567:
4566:
4561:
4552:
4543:
4531:
4530:
4525:
4516:
4515:
4510:
4501:
4492:
4480:
4465:
4463:
4462:
4457:
4449:
4437:
4435:
4434:
4429:
4413:
4411:
4410:
4405:
4403:
4402:
4383:
4381:
4380:
4375:
4370:
4369:
4351:
4350:
4349:
4346:
4343:
4342:
4334:
4320:
4318:
4317:
4312:
4307:
4306:
4301:
4295:
4287:
4282:
4281:
4276:
4264:
4263:
4255:
4245:
4243:
4242:
4237:
4232:
4231:
4226:
4220:
4212:
4207:
4206:
4201:
4179:
4177:
4176:
4171:
4169:
4168:
4163:
4150:
4148:
4147:
4142:
4127:
4125:
4124:
4119:
4087:
4086:
4081:
4068:
4066:
4065:
4060:
4034:
4033:
4028:
4015:
4013:
4012:
4007:
4005:
4004:
3999:
3986:
3984:
3983:
3978:
3976:
3975:
3970:
3957:
3955:
3954:
3949:
3923:
3922:
3917:
3890:
3889:
3884:
3865:
3863:
3862:
3857:
3852:
3851:
3813:
3812:
3805:
3804:
3793:
3792:
3779:
3778:
3767:
3766:
3749:
3748:
3741:
3740:
3729:
3728:
3715:
3714:
3703:
3702:
3675:
3673:
3672:
3667:
3656:
3655:
3650:
3641:
3640:
3635:
3622:
3620:
3619:
3614:
3603:
3602:
3597:
3588:
3587:
3582:
3569:
3567:
3566:
3561:
3550:
3549:
3544:
3535:
3534:
3529:
3516:
3514:
3513:
3508:
3497:
3496:
3491:
3482:
3481:
3476:
3459:
3457:
3456:
3451:
3449:
3448:
3443:
3430:
3428:
3427:
3422:
3420:
3419:
3414:
3401:
3399:
3398:
3393:
3370:
3369:
3364:
3349:
3335:
3321:
3320:
3315:
3299:
3297:
3296:
3291:
3289:
3288:
3283:
3270:
3268:
3267:
3262:
3251:For example, if
3242:
3240:
3239:
3234:
3232:
3231:
3215:
3213:
3212:
3207:
3205:
3204:
3188:
3186:
3185:
3180:
3175:
3174:
3169:
3154:
3153:
3148:
3129:
3127:
3126:
3121:
3116:
3115:
3110:
3089:
3088:
3083:
3065:
3064:
3059:
3038:
3037:
3032:
3020:
3019:
3014:
3002:
3001:
2980:
2979:
2974:
2962:
2961:
2946:
2945:
2940:
2934:
2933:
2915:
2914:
2909:
2903:
2902:
2861:
2859:
2858:
2853:
2851:
2850:
2825:
2823:
2822:
2817:
2815:
2814:
2798:
2796:
2795:
2790:
2785:
2784:
2779:
2764:
2763:
2758:
2742:
2740:
2739:
2734:
2732:
2731:
2715:
2713:
2712:
2707:
2705:
2704:
2699:
2686:
2684:
2683:
2678:
2670:
2669:
2651:
2650:
2638:
2637:
2621:
2619:
2618:
2613:
2601:
2599:
2598:
2593:
2591:
2590:
2572:
2571:
2566:
2560:
2559:
2541:
2540:
2535:
2529:
2528:
2509:
2507:
2506:
2501:
2465:
2463:
2462:
2457:
2455:
2454:
2442:
2441:
2436:
2423:
2421:
2420:
2415:
2404:
2403:
2398:
2377:
2376:
2371:
2362:
2361:
2346:
2345:
2315:
2314:
2309:
2296:
2294:
2293:
2288:
2286:
2285:
2280:
2271:
2270:
2255:
2254:
2233:
2232:
2227:
2218:
2217:
2202:
2201:
2167:
2165:
2164:
2159:
2157:
2156:
2135:
2134:
2129:
2116:
2114:
2113:
2108:
2106:
2105:
2084:
2083:
2078:
2065:
2063:
2062:
2057:
2055:
2054:
2049:
2043:
2042:
2024:
2023:
2018:
2012:
2011:
1989:
1987:
1986:
1981:
1979:
1978:
1973:
1967:
1966:
1948:
1947:
1942:
1936:
1935:
1913:
1911:
1910:
1905:
1881:
1879:
1878:
1873:
1838:
1837:
1832:
1815:
1813:
1812:
1807:
1805:
1804:
1785:
1783:
1782:
1777:
1750:
1749:
1734:
1733:
1728:
1722:
1721:
1703:
1702:
1697:
1691:
1690:
1678:
1677:
1672:
1657:
1655:
1654:
1649:
1644:
1643:
1638:
1623:
1622:
1617:
1601:
1599:
1598:
1593:
1588:
1587:
1582:
1567:
1566:
1561:
1535:
1534:
1522:
1520:
1519:
1514:
1511:
1506:
1487:
1485:
1484:
1479:
1476:
1471:
1456:
1455:
1450:
1441:
1440:
1435:
1419:
1417:
1416:
1411:
1403:
1402:
1383:
1381:
1380:
1375:
1348:
1347:
1332:
1331:
1326:
1320:
1319:
1301:
1300:
1295:
1289:
1288:
1276:
1275:
1270:
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1221:
1220:
1215:
1200:
1199:
1194:
1174:
1172:
1171:
1166:
1164:
1163:
1147:
1145:
1144:
1139:
1127:
1125:
1124:
1119:
1114:
1113:
1108:
1093:
1092:
1087:
1068:
1066:
1065:
1060:
1048:
1046:
1045:
1040:
1038:
1037:
1021:
1019:
1018:
1013:
983:
981:
980:
975:
967:
966:
926:bilinear mapping
922:
920:
919:
914:
873:
871:
870:
865:
824:
822:
821:
816:
802:
800:
799:
794:
781:
779:
778:
773:
771:
770:
753:
751:
750:
745:
715:
713:
712:
707:
705:
704:
685:
683:
682:
677:
658:
656:
655:
650:
631:
629:
628:
623:
621:
620:
589:
587:
586:
581:
579:
575:
571:
445:
443:
442:
437:
424:
422:
421:
416:
414:
413:
397:
395:
394:
389:
350:
348:
347:
342:
314:
312:
311:
306:
304:
289:
287:
286:
281:
279:
278:
262:
260:
259:
254:
252:
251:
232:
230:
229:
224:
209:
207:
206:
201:
151:Early terms for
96:
94:
93:
88:
59:
57:
56:
51:
21:
18091:
18090:
18086:
18085:
18084:
18082:
18081:
18080:
18051:
18050:
18049:
18044:
18028:
18007:
17991:
17970:
17891:
17840:
17830:
17800:
17795:
17777:
17741:Advanced topics
17736:
17660:
17639:
17598:
17564:Hilbert–Schmidt
17537:
17528:Gelfand–Naimark
17475:
17425:
17360:
17346:
17316:
17311:
17293:
17255:
17211:
17148:
17100:
17042:
17033:
16999:Change of basis
16989:Multilinear map
16927:
16909:
16904:
16853:
16835:
16805:
16750:
16722:
16684:
16665:
16634:
16608:
16567:
16541:
16515:
16477:
16455:
16432:
16427:
16426:
16418:
16414:
16406:
16402:
16394:
16390:
16382:
16378:
16370:
16366:
16358:
16354:
16346:
16339:
16331:
16324:
16302:
16298:
16296:
16293:
16292:
16270:
16266:
16265:
16261:
16259:
16256:
16255:
16243:
16239:
16231:
16227:
16219:
16215:
16207:
16203:
16196:
16188:. p. 400.
16182:
16178:
16170:
16166:
16159:
16155:
16147:
16143:
16135:
16131:
16123:
16119:
16111:
16104:
16096:
16092:
16084:
16077:
16072:
16067:
16066:
16053:
16049:
16040:
16036:
16017:
16014:
16013:
15997:
15994:
15993:
15975:
15974:
15969:
15968:
15966:
15963:
15962:
15959:axiom of choice
15956:
15945:
15932:
15924:
15918:
15914:
15890:
15886:
15884:
15881:
15880:
15861:
15859:
15856:
15855:
15838:
15835:
15834:
15814:
15812:
15809:
15808:
15788:
15785:
15784:
15764:
15760:
15758:
15755:
15754:
15737:
15734:
15733:
15715:
15711:
15709:
15706:
15705:
15688:
15684:
15676:
15673:
15672:
15645:
15641:
15639:
15636:
15635:
15633:
15629:
15624:
15581:
15566:
15560:
15507:
15499:
15496:
15495:
15486:
15476:
15470:
15460:
15450:
15444:
15435:
15405:
15372:
15360:
15327:
15264:
15261:
15260:
15251:
15236:
15215:
15212:
15211:
15174:
15170:
15161:
15157:
15127:
15123:
15109:
15100:
15096:
15085:
15082:
15081:
15077:
15070:
15064:
15062:
15055:
15042:
15035:
15028:
15017:
15006:
15000:
14996:
14988:
14985:
14970: /
14966:
14960:
14936:
14932:
14918:
14907:
14904:
14903:
14890:
14868:
14861: /
14856:
14850:
14843: /
14839:
14821: /
14817:
14802: /
14798:
14752:
14734:
14730:
14728:
14725:
14724:
14715:
14700:
14680:
14665:
14650:
14635:
14629:
14610:
14591:
14589:strong topology
14582:
14572:
14562:
14556:
14550:
14544:
14509:
14505:
14494:
14485:
14481:
14472:
14468:
14466:
14463:
14462:
14441:
14434:
14413:
14388:
14365:
14346:
14313:
14302:
14291:
14277:
14274:
14273:
14256:
14250:
14236:
14230:
14219:
14194:
14155:
14153:
14150:
14149:
14136:
14126:
14123:
14113:
14090:anti-isomorphic
14076:
14074:
14051:
14045:
14030:
14021:
14001:
13997:
13991:
13987:
13981:
13966:
13958:
13955:
13954:
13945:
13928:
13925:
13924:
13916:
13904:
13900:
13882:
13877:
13876:
13874:
13871:
13870:
13846:
13841:
13840:
13829:
13826:
13825:
13819:
13800:
13794:
13790:
13779:
13776:
13775:
13755:
13717:
13706:
13701:
13700:
13694:
13690:
13685:
13679:
13668:
13663:
13659:
13658:
13649:
13645:
13640:
13635:
13632:
13631:
13624:
13612:
13596:
13563:
13561:
13558:
13557:
13527:
13525:
13522:
13521:
13511:strong topology
13491:
13489:
13486:
13485:
13458:
13456:
13453:
13452:
13432:
13429:
13428:
13411:
13410:
13408:
13405:
13404:
13388:
13385:
13384:
13365:
13363:
13360:
13359:
13336:
13333:
13332:
13315:
13314:
13312:
13309:
13308:
13292:
13289:
13288:
13265:
13263:
13260:
13259:
13230:
13213:
13195:
13177:
13174:
13173:
13149:
13147:
13144:
13143:
13115:
13112:
13111:
13091:
13088:
13087:
13070:
13069:
13067:
13064:
13063:
13047:
13044:
13043:
13040:bounded subsets
13020:
13018:
13015:
13014:
13011:strong topology
12983:
12982:
12973: for
12971:
12950:
12946:
12922:
12915:
12911:
12896:
12892:
12890:
12887:
12886:
12864:
12862:
12859:
12858:
12833:
12832:
12815:
12808:
12807:
12796:
12789:
12788:
12777:
12775:
12772:
12771:
12752:
12751:
12749:
12746:
12745:
12708:
12701:
12700:
12689:
12682:
12681:
12664:
12662:
12659:
12658:
12639:
12638:
12630:
12627:
12626:
12609:
12608:
12600:
12597:
12596:
12579:
12578:
12570:
12567:
12566:
12534:
12527:
12526:
12515:
12500:
12498:
12495:
12494:
12475:
12474:
12466:
12463:
12462:
12446:
12443:
12442:
12426:
12423:
12422:
12401:
12400:
12398:
12395:
12394:
12364:
12359:
12354:
12324:
12320:
12315:
12303:
12290:
12286:
12274:
12270:
12260:
12259:
12248:
12246:
12243:
12242:
12238:if and only if
12220:
12218:
12215:
12214:
12198:
12195:
12194:
12177:
12173:
12171:
12168:
12167:
12145:
12144:
12142:
12139:
12138:
12122:
12119:
12118:
12102:
12099:
12098:
12082:
12079:
12078:
12056:
12039:
12027:
12014:
12010:
12002:
11999:
11998:
11971:
11970:
11968:
11965:
11964:
11948:
11945:
11944:
11928:
11925:
11924:
11921:bounded subsets
11903:
11902:
11900:
11897:
11896:
11880:
11877:
11876:
11857:
11855:
11852:
11851:
11848:
11840:Main articles:
11838:
11829:
11821:
11810:
11807:
11770:
11769:
11768:
11766:
11763:
11762:
11761:, and its dual
11738:
11737:
11735:
11732:
11731:
11707:
11706:
11705:
11703:
11700:
11699:
11682:
11681:
11679:
11676:
11675:
11647:
11646:
11645:
11643:
11640:
11639:
11622:
11621:
11619:
11616:
11615:
11592:
11591:
11577:
11574:
11573:
11554:
11552:
11549:
11548:
11520:
11517:
11516:
11471:
11469:
11466:
11465:
11448:
11444:
11442:
11439:
11438:
11422:
11420:
11417:
11416:
11400:
11392:
11390:
11387:
11386:
11375:
11340:
11336:
11322:
11319:
11318:
11289:
11284:
11281:
11280:
11278:
11252:
11249:
11248:
11246:
11202:
11199:
11198:
11196:
11169:
11166:
11165:
11163:
11144:
11140:
11138:
11135:
11134:
11132:
11131:-dimensional),
11114:
11111:
11110:
11108:
11090:
11087:
11086:
11084:
11063:
11060:
11059:
11057:
11040:
11036:
11028:
11025:
11024:
10971:
10968:
10967:
10950:
10946:
10938:
10935:
10934:
10912:
10909:
10908:
10905:
10883:
10879:
10877:
10874:
10873:
10856:
10852:
10850:
10847:
10846:
10829:
10825:
10823:
10820:
10819:
10803:
10800:
10799:
10783:
10780:
10779:
10759:
10756:
10755:
10732:
10728:
10719:
10715:
10707:
10699:
10696:
10695:
10676:
10673:
10672:
10652:
10649:
10648:
10647:if and only if
10629:
10624:
10621:
10620:
10592:
10589:
10588:
10587:, a functional
10565:
10560:
10557:
10556:
10537:
10534:
10533:
10517:
10514:
10513:
10486:
10482:
10474:
10471:
10470:
10454:
10451:
10450:
10424:
10420:
10418:
10415:
10414:
10408:vector subspace
10391:
10388:
10387:
10371:
10368:
10367:
10344:
10340:
10331:
10327:
10318:
10314:
10300:
10297:
10296:
10273:
10269:
10260:
10256:
10247:
10243:
10229:
10226:
10225:
10206:
10203:
10202:
10186:
10183:
10182:
10166:
10163:
10162:
10139:
10134:
10118:
10105:
10094:
10090:
10078:
10073:
10069:
10068:
10066:
10063:
10062:
10043:
10040:
10039:
10023:
10020:
10019:
10003:
10000:
9999:
9976:
9972:
9966:
9962:
9957:
9954:
9953:
9930:
9926:
9905:
9901:
9892:
9888:
9886:
9883:
9882:
9863:
9860:
9859:
9843:
9840:
9839:
9823:
9820:
9819:
9796:
9792:
9783:
9779:
9770:
9766:
9752:
9749:
9748:
9705:
9702:
9701:
9685:
9682:
9681:
9664:
9660:
9639:
9635:
9633:
9630:
9629:
9612:
9608:
9599:
9595:
9587:
9584:
9583:
9553:
9548:
9547:
9542:
9539:
9538:
9522:
9519:
9518:
9490:
9487:
9486:
9469:
9465:
9463:
9460:
9459:
9437:
9434:
9433:
9399:
9396:
9395:
9378:
9374:
9366:
9363:
9362:
9345:
9341:
9339:
9336:
9335:
9334:, denoted here
9318:
9314:
9312:
9309:
9308:
9292:
9289:
9288:
9266:
9263:
9262:
9261:be a subset of
9246:
9243:
9242:
9239:
9157:category theory
9142:
9120:
9115:if and only if
9082:
9081:The assignment
8997:
8993:
8988:
8985:
8984:
8958:
8955:
8954:
8935:
8931:
8923:
8920:
8919:
8898:
8895:
8894:
8876:
8873:
8872:
8847:
8843:
8841:
8838:
8837:
8809:
8805:
8803:
8800:
8799:
8780:
8776:
8768:
8765:
8764:
8721:
8717:
8715:
8712:
8711:
8694:
8666:
8663:
8657:
8632:
8629:
8628:
8604:
8601:
8600:
8583:
8575:
8574:
8566:
8563:
8562:
8542:
8538:
8524:
8521:
8520:
8489:
8486:
8485:
8462:
8458:
8449:
8441:
8440:
8438:
8435:
8434:
8417:
8413:
8393:
8390:
8389:
8334:
8331:
8330:
8296:
8275:
8271:
8250:
8246:
8244:
8241:
8240:
8224:
8221:
8220:
8204:
8201:
8200:
8191:
8150:
8130:
8127:
8126:
8109:
8087:
8083:
8081:
8079:
8076:
8075:
8047:
8043:
8041:
8014:
8010:
8008:
8005:
8004:
7989:
7881:
7877:
7863:
7860:
7859:
7834:
7830:
7816:
7813:
7812:
7811:
7802:(resp., all of
7779:
7776:
7775:
7686:
7682:
7681:
7677:
7675:
7672:
7671:
7648:
7644:
7617:
7613:
7611:
7608:
7607:
7592:
7582:
7541:
7538:
7537:
7522:
7500:
7468:
7459:
7455:
7441:
7436:
7428:
7420:
7418:
7415:
7414:
7391:
7387:
7373:
7350:
7348:
7345:
7344:
7341:absolute values
7315:
7307:
7299:
7290:
7286:
7272:
7267:
7253:
7245:
7239:
7235:
7230:
7220:
7212:
7211:
7206:
7205:
7197:
7189:
7181:
7158:
7156:
7153:
7152:
7129:
7126:
7125:
7109:
7106:
7105:
7092:cardinal number
7056:
7052:
7034:
7021:
7017:
7005:
6992:
6972:
6967:
6963:
6962:
6953:
6949:
6947:
6944:
6943:
6924:
6921:
6920:
6904:
6901:
6900:
6879:
6875:
6873:
6870:
6869:
6837:
6824:
6820:
6814:
6810:
6799:
6796:
6795:
6776:
6773:
6772:
6756:
6753:
6752:
6731:
6727:
6721:
6717:
6712:
6709:
6708:
6689:
6686:
6685:
6668:
6664:
6662:
6659:
6658:
6635:
6631:
6625:
6621:
6609:
6593:
6589:
6577:
6573:
6561:
6543:
6538:
6537:
6531:
6527:
6515:
6510:
6506:
6501:
6498:
6497:
6478:
6475:
6474:
6458:
6455:
6454:
6437:
6433:
6416:
6413:
6412:
6384:
6381:
6380:
6364:
6361:
6360:
6340:
6335:
6334:
6319:
6315:
6313:
6310:
6309:
6293:
6290:
6289:
6273:
6270:
6269:
6253:
6250:
6249:
6233:
6230:
6229:
6228:functions from
6208:
6204:
6202:
6199:
6198:
6182:
6179:
6178:
6153:
6150:
6149:
6133:
6130:
6129:
6113:
6110:
6109:
6089:
6084:
6083:
6077:
6073:
6061:
6055:
6052:
6051:
6032:
6029:
6028:
6006:
6003:
6002:
5970:
5966:
5964:
5961:
5960:
5932:
5929:
5928:
5911:
5907:
5901:
5897:
5892:
5889:
5888:
5887:with the space
5872:
5869:
5868:
5836:
5831:
5830:
5825:
5822:
5821:
5805:
5802:
5801:
5800:over any field
5785:
5782:
5781:
5760:
5759:
5754:
5753:
5751:
5748:
5747:
5726:
5721:
5720:
5718:
5715:
5714:
5693:
5689:
5687:
5684:
5683:
5660:
5656:
5650:
5646:
5640:
5634:
5631:
5630:
5610:
5605:
5604:
5602:
5599:
5598:
5578:
5574:
5569:
5566:
5565:
5545:
5541:
5536:
5533:
5532:
5527:, the space of
5510:
5509:
5504:
5503:
5501:
5498:
5497:
5480:
5475:
5474:
5472:
5469:
5468:
5452:
5449:
5448:
5431:
5426:
5425:
5423:
5420:
5419:
5403:
5395:
5392:
5391:
5375:
5373:
5370:
5369:
5348:
5343:
5342:
5340:
5337:
5336:
5311:
5308:
5307:
5290:
5285:
5284:
5282:
5279:
5278:
5258:
5255:
5254:
5237:
5232:
5231:
5229:
5226:
5225:
5206:
5203:
5202:
5199:
5178:
5175:
5174:
5157:
5153:
5151:
5148:
5147:
5127:
5124:
5123:
5105:
5101:
5099:
5096:
5095:
5079:
5076:
5075:
5058:
5054:
5052:
5049:
5048:
5024:
5021:
5020:
5001:
4998:
4997:
4975:
4972:
4971:
4955:
4952:
4951:
4926:
4923:
4922:
4900:
4897:
4896:
4880:
4877:
4876:
4854:
4851:
4850:
4834:
4831:
4830:
4814:
4811:
4810:
4794:
4791:
4790:
4774:
4771:
4770:
4754:
4751:
4750:
4729:
4724:
4723:
4721:
4718:
4717:
4701:
4698:
4697:
4674:
4671:
4670:
4653:
4648:
4647:
4645:
4642:
4641:
4640:In particular,
4604:
4601:
4600:
4577:
4572:
4571:
4562:
4557:
4556:
4548:
4539:
4526:
4521:
4520:
4511:
4506:
4505:
4497:
4488:
4476:
4474:
4471:
4470:
4445:
4443:
4440:
4439:
4423:
4420:
4419:
4416:identity matrix
4398:
4394:
4392:
4389:
4388:
4365:
4361:
4345:
4344:
4333:
4332:
4331:
4329:
4326:
4325:
4302:
4297:
4296:
4291:
4283:
4277:
4272:
4271:
4254:
4253:
4251:
4248:
4247:
4227:
4222:
4221:
4216:
4208:
4202:
4197:
4196:
4185:
4182:
4181:
4164:
4159:
4158:
4156:
4153:
4152:
4136:
4133:
4132:
4082:
4077:
4076:
4074:
4071:
4070:
4029:
4024:
4023:
4021:
4018:
4017:
4000:
3995:
3994:
3992:
3989:
3988:
3971:
3966:
3965:
3963:
3960:
3959:
3918:
3913:
3912:
3885:
3880:
3879:
3874:
3871:
3870:
3846:
3845:
3840:
3834:
3833:
3828:
3818:
3817:
3807:
3806:
3800:
3796:
3794:
3788:
3784:
3781:
3780:
3774:
3770:
3768:
3762:
3758:
3751:
3750:
3743:
3742:
3736:
3732:
3730:
3724:
3720:
3717:
3716:
3710:
3706:
3704:
3698:
3694:
3687:
3686:
3684:
3681:
3680:
3651:
3646:
3645:
3636:
3631:
3630:
3628:
3625:
3624:
3598:
3593:
3592:
3583:
3578:
3577:
3575:
3572:
3571:
3545:
3540:
3539:
3530:
3525:
3524:
3522:
3519:
3518:
3492:
3487:
3486:
3477:
3472:
3471:
3469:
3466:
3465:
3444:
3439:
3438:
3436:
3433:
3432:
3415:
3410:
3409:
3407:
3404:
3403:
3365:
3360:
3359:
3345:
3331:
3316:
3311:
3310:
3305:
3302:
3301:
3284:
3279:
3278:
3276:
3273:
3272:
3256:
3253:
3252:
3249:
3227:
3223:
3221:
3218:
3217:
3200:
3196:
3194:
3191:
3190:
3170:
3165:
3164:
3149:
3144:
3143:
3138:
3135:
3134:
3111:
3106:
3105:
3084:
3079:
3078:
3060:
3055:
3054:
3033:
3028:
3027:
3015:
3010:
3009:
2997:
2993:
2975:
2970:
2969:
2957:
2953:
2941:
2936:
2935:
2929:
2925:
2910:
2905:
2904:
2898:
2894:
2871:
2868:
2867:
2846:
2842:
2834:
2831:
2830:
2810:
2806:
2804:
2801:
2800:
2780:
2775:
2774:
2759:
2754:
2753:
2748:
2745:
2744:
2727:
2723:
2721:
2718:
2717:
2700:
2695:
2694:
2692:
2689:
2688:
2665:
2661:
2646:
2642:
2633:
2629:
2627:
2624:
2623:
2607:
2604:
2603:
2586:
2582:
2567:
2562:
2561:
2555:
2551:
2536:
2531:
2530:
2524:
2520:
2518:
2515:
2514:
2471:
2468:
2467:
2450:
2446:
2437:
2432:
2431:
2429:
2426:
2425:
2399:
2394:
2393:
2372:
2367:
2366:
2357:
2353:
2341:
2337:
2310:
2305:
2304:
2302:
2299:
2298:
2281:
2276:
2275:
2266:
2262:
2250:
2246:
2228:
2223:
2222:
2213:
2209:
2197:
2193:
2173:
2170:
2169:
2152:
2148:
2130:
2125:
2124:
2122:
2119:
2118:
2101:
2097:
2079:
2074:
2073:
2071:
2068:
2067:
2050:
2045:
2044:
2038:
2034:
2019:
2014:
2013:
2007:
2003:
1995:
1992:
1991:
1974:
1969:
1968:
1962:
1958:
1943:
1938:
1937:
1931:
1927:
1919:
1916:
1915:
1887:
1884:
1883:
1833:
1828:
1827:
1825:
1822:
1821:
1800:
1796:
1794:
1791:
1790:
1745:
1741:
1729:
1724:
1723:
1717:
1713:
1698:
1693:
1692:
1686:
1682:
1673:
1668:
1667:
1665:
1662:
1661:
1639:
1634:
1633:
1618:
1613:
1612:
1607:
1604:
1603:
1583:
1578:
1577:
1562:
1557:
1556:
1551:
1548:
1547:
1540:
1525:Kronecker delta
1507:
1502:
1496:
1493:
1492:
1472:
1467:
1451:
1446:
1445:
1436:
1431:
1430:
1428:
1425:
1424:
1398:
1394:
1392:
1389:
1388:
1343:
1339:
1327:
1322:
1321:
1315:
1311:
1296:
1291:
1290:
1284:
1280:
1271:
1266:
1265:
1263:
1260:
1259:
1240:
1237:
1236:
1216:
1211:
1210:
1195:
1190:
1189:
1184:
1181:
1180:
1159:
1155:
1153:
1150:
1149:
1133:
1130:
1129:
1109:
1104:
1103:
1088:
1083:
1082:
1077:
1074:
1073:
1054:
1051:
1050:
1033:
1029:
1027:
1024:
1023:
1007:
1004:
1003:
1000:
994:
986:natural pairing
962:
958:
932:
929:
928:
881:
878:
877:
832:
829:
828:
810:
807:
806:
788:
785:
784:
782:and an element
766:
762:
760:
757:
756:
739:
736:
735:
700:
696:
694:
691:
690:
665:
662:
661:
638:
635:
634:
616:
612:
598:
595:
594:
577:
576:
558:
554:
544:
520:
519:
485:
457:
455:
452:
451:
431:
428:
427:
409:
405:
403:
400:
399:
365:
362:
361:
324:
321:
320:
297:
295:
292:
291:
274:
270:
268:
265:
264:
247:
243:
241:
238:
237:
218:
215:
214:
195:
192:
191:
185:
171:and . The term
161:espace conjugué
79:
76:
75:
45:
42:
41:
28:
23:
22:
15:
12:
11:
5:
18089:
18079:
18078:
18073:
18068:
18066:Linear algebra
18063:
18046:
18045:
18043:
18042:
18036:
18034:
18033:Other concepts
18030:
18029:
18027:
18026:
18021:
18015:
18013:
18009:
18008:
18006:
18005:
17999:
17997:
17993:
17992:
17990:
17989:
17984:
17982:Banach–Alaoglu
17978:
17976:
17972:
17971:
17969:
17968:
17963:
17962:
17961:
17956:
17954:polar topology
17946:
17941:
17940:
17939:
17934:
17929:
17919:
17914:
17913:
17912:
17901:
17899:
17893:
17892:
17890:
17889:
17884:
17882:Polar topology
17879:
17874:
17869:
17864:
17859:
17854:
17848:
17846:
17845:Basic concepts
17842:
17841:
17835:and spaces of
17829:
17828:
17821:
17814:
17806:
17797:
17796:
17794:
17793:
17782:
17779:
17778:
17776:
17775:
17770:
17765:
17760:
17758:Choquet theory
17755:
17750:
17744:
17742:
17738:
17737:
17735:
17734:
17724:
17719:
17714:
17709:
17704:
17699:
17694:
17689:
17684:
17679:
17674:
17668:
17666:
17662:
17661:
17659:
17658:
17653:
17647:
17645:
17641:
17640:
17638:
17637:
17632:
17627:
17622:
17617:
17612:
17610:Banach algebra
17606:
17604:
17600:
17599:
17597:
17596:
17591:
17586:
17581:
17576:
17571:
17566:
17561:
17556:
17551:
17545:
17543:
17539:
17538:
17536:
17535:
17533:Banach–Alaoglu
17530:
17525:
17520:
17515:
17510:
17505:
17500:
17495:
17489:
17487:
17481:
17480:
17477:
17476:
17474:
17473:
17468:
17463:
17461:Locally convex
17458:
17444:
17439:
17433:
17431:
17427:
17426:
17424:
17423:
17418:
17413:
17408:
17403:
17398:
17393:
17388:
17383:
17378:
17372:
17366:
17362:
17361:
17345:
17344:
17337:
17330:
17322:
17313:
17312:
17310:
17309:
17298:
17295:
17294:
17292:
17291:
17286:
17281:
17276:
17271:
17269:Floating-point
17265:
17263:
17257:
17256:
17254:
17253:
17251:Tensor product
17248:
17243:
17238:
17236:Function space
17233:
17228:
17222:
17220:
17213:
17212:
17210:
17209:
17204:
17199:
17194:
17189:
17184:
17179:
17174:
17172:Triple product
17169:
17164:
17158:
17156:
17150:
17149:
17147:
17146:
17141:
17136:
17131:
17126:
17121:
17116:
17110:
17108:
17102:
17101:
17099:
17098:
17093:
17088:
17086:Transformation
17083:
17078:
17076:Multiplication
17073:
17068:
17063:
17058:
17052:
17050:
17044:
17043:
17036:
17034:
17032:
17031:
17026:
17021:
17016:
17011:
17006:
17001:
16996:
16991:
16986:
16981:
16976:
16971:
16966:
16961:
16956:
16951:
16946:
16941:
16935:
16933:
16932:Basic concepts
16929:
16928:
16926:
16925:
16920:
16914:
16911:
16910:
16907:Linear algebra
16903:
16902:
16895:
16888:
16880:
16874:
16873:
16852:
16851:External links
16849:
16848:
16847:
16833:
16817:
16803:
16783:
16771:
16762:
16748:
16726:
16720:
16696:
16683:978-1584888666
16682:
16669:
16663:
16647:Thorne, Kip S.
16639:
16632:
16612:
16606:
16586:
16565:
16545:
16539:
16519:
16513:
16493:
16481:
16475:
16459:
16453:
16431:
16428:
16425:
16424:
16412:
16400:
16388:
16376:
16364:
16352:
16337:
16322:
16305:
16301:
16276:
16273:
16269:
16264:
16248:(2012-12-29).
16237:
16225:
16213:
16201:
16194:
16176:
16164:
16153:
16141:
16129:
16117:
16102:
16090:
16074:
16073:
16071:
16068:
16065:
16064:
16047:
16034:
16021:
16001:
15978:
15972:
15943:
15912:
15910:, p. 35).
15893:
15889:
15867:
15864:
15842:
15820:
15817:
15792:
15772:
15767:
15763:
15741:
15718:
15714:
15691:
15687:
15683:
15680:
15648:
15644:
15626:
15625:
15623:
15620:
15619:
15618:
15612:
15607:
15602:
15597:
15592:
15587:
15580:
15577:
15557:
15556:
15545:
15542:
15539:
15536:
15532:
15529:
15526:
15523:
15520:
15517:
15513:
15510:
15506:
15503:
15420:locally convex
15349:
15348:
15337:
15333:
15330:
15326:
15323:
15317:
15314:
15311:
15308:
15304:
15301:
15298:
15295:
15292:
15289:
15286:
15283:
15280:
15277:
15274:
15271:
15268:
15219:
15197:
15194:
15191:
15188:
15185:
15182:
15177:
15173:
15169:
15164:
15160:
15156:
15153:
15150:
15147:
15144:
15141:
15138:
15135:
15130:
15126:
15122:
15119:
15115:
15112:
15108:
15103:
15099:
15095:
15092:
15089:
15075:
15068:
15060:
15053:
15040:
15033:
15016:
15013:
14984:
14981:
14953:
14952:
14939:
14935:
14931:
14928:
14924:
14921:
14917:
14914:
14911:
14826:. Indeed, let
14795:
14794:
14783:
14780:
14777:
14774:
14771:
14768:
14765:
14762:
14758:
14755:
14751:
14748:
14745:
14742:
14737:
14733:
14699:
14696:
14529:
14528:
14517:
14512:
14508:
14504:
14500:
14497:
14493:
14488:
14484:
14480:
14475:
14471:
14432:
14393:is injective.
14335:
14334:
14323:
14319:
14316:
14312:
14308:
14305:
14301:
14297:
14294:
14290:
14287:
14284:
14281:
14216:
14215:
14204:
14200:
14197:
14193:
14190:
14186:
14183:
14180:
14177:
14174:
14171:
14168:
14165:
14161:
14158:
14112:
14109:
14072:
14028:
14018:
14017:
14004:
14000:
13994:
13990:
13984:
13980:
13976:
13973:
13969:
13965:
13962:
13950:is defined by
13932:
13912:
13885:
13880:
13857:
13854:
13849:
13844:
13839:
13836:
13833:
13806:
13803:
13797:
13793:
13789:
13786:
13783:
13748:
13747:
13736:
13733:
13730:
13724:
13721:
13715:
13709:
13704:
13697:
13693:
13688:
13682:
13677:
13674:
13671:
13667:
13662:
13657:
13652:
13648:
13643:
13639:
13620:
13595:
13592:
13591:
13590:
13569:
13566:
13554:
13533:
13530:
13518:
13497:
13494:
13464:
13461:
13449:
13448:
13436:
13414:
13392:
13371:
13368:
13352:
13340:
13318:
13296:
13271:
13268:
13251:
13250:
13249:
13248:
13237:
13233:
13229:
13226:
13223:
13220:
13216:
13210:
13207:
13204:
13201:
13198:
13194:
13190:
13187:
13184:
13181:
13155:
13152:
13119:
13108:
13107:
13095:
13073:
13051:
13026:
13023:
13000:
12999:
12986:
12981:
12978:
12969:
12965:
12961:
12958:
12953:
12949:
12945:
12942:
12935:
12928:
12925:
12921:
12918:
12914:
12907:
12899:
12895:
12870:
12867:
12855:
12854:
12853:
12852:
12841:
12836:
12831:
12828:
12825:
12822:
12811:
12806:
12803:
12792:
12787:
12784:
12755:
12743:
12742:
12741:
12730:
12727:
12724:
12721:
12718:
12715:
12704:
12699:
12696:
12685:
12680:
12677:
12674:
12671:
12642:
12637:
12634:
12612:
12607:
12604:
12582:
12577:
12574:
12565:Each two sets
12563:
12562:
12561:
12550:
12547:
12544:
12541:
12530:
12525:
12522:
12513:
12510:
12507:
12478:
12473:
12470:
12450:
12430:
12404:
12391:
12390:
12379:
12373:
12370:
12367:
12363:
12357:
12353:
12350:
12347:
12344:
12341:
12338:
12335:
12332:
12327:
12323:
12318:
12312:
12309:
12306:
12302:
12298:
12293:
12289:
12285:
12282:
12277:
12273:
12269:
12263:
12258:
12255:
12226:
12223:
12202:
12180:
12176:
12153:
12148:
12126:
12106:
12086:
12075:
12074:
12063:
12059:
12055:
12052:
12049:
12046:
12042:
12036:
12033:
12030:
12026:
12022:
12017:
12013:
12009:
12006:
11979:
11974:
11952:
11932:
11906:
11884:
11863:
11860:
11842:Polar topology
11837:
11834:
11806:
11803:
11782:
11778:
11773:
11759:Schwartz space
11746:
11741:
11719:
11715:
11710:
11685:
11659:
11655:
11650:
11625:
11613:test functions
11595:
11590:
11587:
11584:
11581:
11560:
11557:
11524:
11477:
11474:
11451:
11447:
11425:
11403:
11399:
11395:
11374:
11371:
11367:inverse length
11354:
11351:
11346:
11343:
11339:
11335:
11332:
11329:
11326:
11315:inverse second
11296:
11292:
11288:
11273:: occurrences
11256:
11206:
11182:
11179:
11176:
11173:
11162:behaves as an
11147:
11143:
11118:
11094:
11070:
11067:
11043:
11039:
11035:
11032:
11008:
11005:
11002:
10999:
10996:
10993:
10990:
10987:
10984:
10981:
10978:
10975:
10953:
10949:
10945:
10942:
10922:
10919:
10916:
10904:
10901:
10886:
10882:
10859:
10855:
10832:
10828:
10807:
10787:
10763:
10752:
10751:
10740:
10735:
10731:
10727:
10722:
10718:
10714:
10710:
10706:
10703:
10680:
10656:
10636:
10632:
10628:
10608:
10605:
10602:
10599:
10596:
10572:
10568:
10564:
10554:quotient space
10541:
10521:
10492:
10489:
10485:
10481:
10478:
10458:
10447:
10446:
10435:
10432:
10427:
10423:
10395:
10375:
10364:
10363:
10352:
10347:
10343:
10339:
10334:
10330:
10326:
10321:
10317:
10313:
10310:
10307:
10304:
10290:
10289:
10276:
10272:
10268:
10263:
10259:
10255:
10250:
10246:
10242:
10239:
10236:
10233:
10210:
10190:
10170:
10159:
10158:
10147:
10142:
10137:
10133:
10127:
10124:
10121:
10117:
10113:
10108:
10103:
10097:
10093:
10087:
10084:
10081:
10077:
10072:
10047:
10027:
10007:
9985:
9982:
9979:
9975:
9969:
9965:
9961:
9950:
9949:
9938:
9933:
9929:
9925:
9922:
9919:
9916:
9913:
9908:
9904:
9900:
9895:
9891:
9867:
9847:
9827:
9816:
9815:
9804:
9799:
9795:
9791:
9786:
9782:
9778:
9773:
9769:
9765:
9762:
9759:
9756:
9733:
9730:
9727:
9724:
9721:
9718:
9715:
9712:
9709:
9689:
9667:
9663:
9659:
9656:
9653:
9650:
9647:
9642:
9638:
9615:
9611:
9607:
9602:
9598:
9594:
9591:
9564:
9561:
9556:
9551:
9546:
9526:
9506:
9503:
9500:
9497:
9494:
9472:
9468:
9447:
9444:
9441:
9421:
9418:
9415:
9412:
9409:
9406:
9403:
9381:
9377:
9373:
9370:
9348:
9344:
9321:
9317:
9296:
9270:
9250:
9238:
9235:
9064:
9063:
9052:
9049:
9046:
9043:
9040:
9037:
9033:
9030:
9027:
9024:
9021:
9018:
9014:
9011:
9008:
9005:
9000:
8996:
8992:
8968:
8965:
8962:
8938:
8934:
8930:
8927:
8902:
8880:
8864:is called the
8850:
8846:
8823:
8820:
8817:
8812:
8808:
8783:
8779:
8775:
8772:
8760:
8759:
8747:
8744:
8741:
8738:
8735:
8732:
8729:
8724:
8720:
8707:is defined by
8659:Main article:
8656:
8653:
8636:
8608:
8586:
8581:
8578:
8573:
8570:
8548:
8545:
8541:
8537:
8534:
8531:
8528:
8508:
8505:
8502:
8499:
8496:
8493:
8473:
8470:
8465:
8461:
8457:
8452:
8447:
8444:
8420:
8416:
8412:
8409:
8406:
8403:
8400:
8397:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8341:
8338:
8318:
8314:
8311:
8308:
8305:
8302:
8299:
8292:
8289:
8286:
8283:
8278:
8274:
8270:
8267:
8264:
8261:
8256:
8253:
8249:
8228:
8208:
8190:
8187:
8186:
8185:
8174:
8171:
8168:
8165:
8162:
8157:
8154:
8149:
8146:
8143:
8140:
8137:
8134:
8095:
8090:
8086:
8072:
8071:
8060:
8055:
8050:
8046:
8040:
8037:
8034:
8029:
8026:
8023:
8020:
8017:
8013:
7959:
7958:
7946:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7919:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7895:
7892:
7889:
7884:
7880:
7876:
7873:
7870:
7867:
7837:
7833:
7829:
7826:
7823:
7820:
7783:
7756:
7755:
7744:
7741:
7738:
7735:
7732:
7729:
7726:
7722:
7718:
7715:
7712:
7709:
7706:
7701:
7698:
7695:
7692:
7689:
7685:
7680:
7665:
7664:
7651:
7647:
7643:
7640:
7637:
7632:
7629:
7626:
7623:
7620:
7616:
7575:
7574:
7563:
7560:
7557:
7554:
7551:
7548:
7545:
7499:
7496:
7475:
7471:
7467:
7462:
7458:
7454:
7450:
7447:
7444:
7439:
7435:
7431:
7427:
7423:
7402:
7399:
7394:
7390:
7386:
7382:
7379:
7376:
7372:
7369:
7366:
7363:
7359:
7356:
7353:
7337:
7336:
7325:
7322:
7318:
7314:
7310:
7306:
7302:
7298:
7293:
7289:
7285:
7281:
7278:
7275:
7270:
7266:
7262:
7259:
7256:
7252:
7248:
7242:
7238:
7233:
7229:
7223:
7219:
7215:
7209:
7204:
7200:
7196:
7192:
7188:
7184:
7180:
7177:
7174:
7171:
7167:
7164:
7161:
7148:implies that
7133:
7113:
7077:general result
7073:
7072:
7059:
7055:
7051:
7048:
7043:
7040:
7037:
7033:
7029:
7024:
7020:
7014:
7011:
7008:
7004:
7000:
6995:
6990:
6986:
6981:
6978:
6975:
6971:
6966:
6961:
6956:
6952:
6928:
6908:
6897:direct product
6882:
6878:
6866:
6865:
6854:
6851:
6846:
6843:
6840:
6836:
6832:
6827:
6823:
6817:
6813:
6809:
6806:
6803:
6780:
6760:
6734:
6730:
6724:
6720:
6716:
6693:
6671:
6667:
6655:
6654:
6643:
6638:
6634:
6628:
6624:
6618:
6615:
6612:
6608:
6604:
6601:
6596:
6592:
6588:
6585:
6580:
6576:
6570:
6567:
6564:
6560:
6556:
6552:
6546:
6541:
6534:
6530:
6524:
6521:
6518:
6514:
6509:
6505:
6482:
6462:
6440:
6436:
6432:
6429:
6426:
6423:
6420:
6400:
6397:
6394:
6391:
6388:
6368:
6348:
6343:
6338:
6333:
6330:
6327:
6322:
6318:
6297:
6277:
6257:
6237:
6211:
6207:
6186:
6163:
6160:
6157:
6137:
6117:
6106:
6105:
6092:
6087:
6080:
6076:
6070:
6067:
6064:
6060:
6036:
6016:
6013:
6010:
5990:
5987:
5984:
5981:
5978:
5973:
5969:
5948:
5945:
5942:
5939:
5936:
5914:
5910:
5904:
5900:
5896:
5876:
5856:
5853:
5850:
5847:
5844:
5839:
5834:
5829:
5809:
5789:
5763:
5757:
5729:
5724:
5696:
5692:
5680:
5679:
5668:
5663:
5659:
5653:
5649:
5643:
5639:
5613:
5608:
5586:
5581:
5577:
5573:
5553:
5548:
5544:
5540:
5513:
5507:
5483:
5478:
5456:
5434:
5429:
5406:
5402:
5399:
5378:
5351:
5346:
5321:
5318:
5315:
5293:
5288:
5262:
5240:
5235:
5210:
5198:
5195:
5182:
5160:
5156:
5131:
5108:
5104:
5083:
5061:
5057:
5028:
5005:
4985:
4982:
4979:
4959:
4939:
4936:
4933:
4930:
4910:
4907:
4904:
4884:
4864:
4861:
4858:
4838:
4818:
4798:
4778:
4758:
4732:
4727:
4705:
4678:
4656:
4651:
4620:
4617:
4614:
4611:
4608:
4597:
4596:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4551:
4547:
4542:
4538:
4534:
4529:
4524:
4519:
4514:
4509:
4504:
4500:
4496:
4491:
4487:
4483:
4479:
4455:
4452:
4448:
4427:
4401:
4397:
4385:
4384:
4373:
4368:
4364:
4360:
4357:
4354:
4340:
4337:
4310:
4305:
4300:
4294:
4290:
4286:
4280:
4275:
4270:
4267:
4261:
4258:
4235:
4230:
4225:
4219:
4215:
4211:
4205:
4200:
4195:
4192:
4189:
4167:
4162:
4140:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4085:
4080:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4032:
4027:
4003:
3998:
3974:
3969:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3921:
3916:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3888:
3883:
3878:
3867:
3866:
3855:
3850:
3844:
3841:
3839:
3836:
3835:
3832:
3829:
3827:
3824:
3823:
3821:
3816:
3811:
3803:
3799:
3795:
3791:
3787:
3783:
3782:
3777:
3773:
3769:
3765:
3761:
3757:
3756:
3754:
3747:
3739:
3735:
3731:
3727:
3723:
3719:
3718:
3713:
3709:
3705:
3701:
3697:
3693:
3692:
3690:
3665:
3662:
3659:
3654:
3649:
3644:
3639:
3634:
3612:
3609:
3606:
3601:
3596:
3591:
3586:
3581:
3559:
3556:
3553:
3548:
3543:
3538:
3533:
3528:
3506:
3503:
3500:
3495:
3490:
3485:
3480:
3475:
3447:
3442:
3418:
3413:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3368:
3363:
3358:
3355:
3352:
3348:
3344:
3341:
3338:
3334:
3330:
3327:
3324:
3319:
3314:
3309:
3287:
3282:
3260:
3246:
3245:
3230:
3226:
3203:
3199:
3178:
3173:
3168:
3163:
3160:
3157:
3152:
3147:
3142:
3131:
3130:
3119:
3114:
3109:
3104:
3101:
3098:
3095:
3092:
3087:
3082:
3077:
3074:
3071:
3068:
3063:
3058:
3053:
3050:
3047:
3044:
3041:
3036:
3031:
3026:
3023:
3018:
3013:
3008:
3005:
3000:
2996:
2992:
2989:
2986:
2983:
2978:
2973:
2968:
2965:
2960:
2956:
2952:
2949:
2944:
2939:
2932:
2928:
2924:
2921:
2918:
2913:
2908:
2901:
2897:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2864:
2863:
2849:
2845:
2841:
2838:
2827:
2813:
2809:
2788:
2783:
2778:
2773:
2770:
2767:
2762:
2757:
2752:
2743:). Therefore,
2730:
2726:
2703:
2698:
2676:
2673:
2668:
2664:
2660:
2657:
2654:
2649:
2645:
2641:
2636:
2632:
2611:
2589:
2585:
2581:
2578:
2575:
2570:
2565:
2558:
2554:
2550:
2547:
2544:
2539:
2534:
2527:
2523:
2511:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2453:
2449:
2445:
2440:
2435:
2413:
2410:
2407:
2402:
2397:
2392:
2389:
2386:
2383:
2380:
2375:
2370:
2365:
2360:
2356:
2352:
2349:
2344:
2340:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2313:
2308:
2284:
2279:
2274:
2269:
2265:
2261:
2258:
2253:
2249:
2245:
2242:
2239:
2236:
2231:
2226:
2221:
2216:
2212:
2208:
2205:
2200:
2196:
2192:
2189:
2186:
2183:
2180:
2177:
2155:
2151:
2147:
2144:
2141:
2138:
2133:
2128:
2104:
2100:
2096:
2093:
2090:
2087:
2082:
2077:
2053:
2048:
2041:
2037:
2033:
2030:
2027:
2022:
2017:
2010:
2006:
2002:
1999:
1977:
1972:
1965:
1961:
1957:
1954:
1951:
1946:
1941:
1934:
1930:
1926:
1923:
1903:
1900:
1897:
1894:
1891:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1836:
1831:
1803:
1799:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1753:
1748:
1744:
1740:
1737:
1732:
1727:
1720:
1716:
1712:
1709:
1706:
1701:
1696:
1689:
1685:
1681:
1676:
1671:
1647:
1642:
1637:
1632:
1629:
1626:
1621:
1616:
1611:
1591:
1586:
1581:
1576:
1573:
1570:
1565:
1560:
1555:
1542:
1541:
1538:
1533:
1510:
1505:
1501:
1489:
1488:
1475:
1470:
1466:
1462:
1459:
1454:
1449:
1444:
1439:
1434:
1409:
1406:
1401:
1397:
1385:
1384:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1351:
1346:
1342:
1338:
1335:
1330:
1325:
1318:
1314:
1310:
1307:
1304:
1299:
1294:
1287:
1283:
1279:
1274:
1269:
1244:
1224:
1219:
1214:
1209:
1206:
1203:
1198:
1193:
1188:
1162:
1158:
1137:
1117:
1112:
1107:
1102:
1099:
1096:
1091:
1086:
1081:
1058:
1036:
1032:
1011:
993:
990:
973:
970:
965:
961:
957:
954:
951:
948:
945:
942:
939:
936:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
814:
792:
769:
765:
743:
703:
699:
675:
672:
669:
648:
645:
642:
619:
615:
611:
608:
605:
602:
591:
590:
574:
570:
567:
564:
561:
557:
553:
550:
547:
545:
543:
540:
537:
534:
531:
528:
525:
522:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
486:
484:
481:
478:
475:
472:
469:
466:
463:
460:
459:
435:
412:
408:
387:
384:
381:
378:
375:
372:
369:
340:
337:
334:
331:
328:
303:
300:
277:
273:
250:
246:
222:
199:
184:
181:
142:Hilbert spaces
128:analysis with
108:
86:
83:
49:
26:
9:
6:
4:
3:
2:
18088:
18077:
18074:
18072:
18069:
18067:
18064:
18062:
18059:
18058:
18056:
18041:
18038:
18037:
18035:
18031:
18025:
18022:
18020:
18017:
18016:
18014:
18010:
18004:
18001:
18000:
17998:
17994:
17988:
17985:
17983:
17980:
17979:
17977:
17973:
17967:
17964:
17960:
17957:
17955:
17952:
17951:
17950:
17947:
17945:
17942:
17938:
17935:
17933:
17930:
17928:
17925:
17924:
17923:
17920:
17918:
17915:
17911:
17908:
17907:
17906:
17905:Norm topology
17903:
17902:
17900:
17898:
17894:
17888:
17885:
17883:
17880:
17878:
17875:
17873:
17870:
17868:
17865:
17863:
17862:Dual topology
17860:
17858:
17855:
17853:
17850:
17849:
17847:
17843:
17838:
17834:
17827:
17822:
17820:
17815:
17813:
17808:
17807:
17804:
17792:
17784:
17783:
17780:
17774:
17771:
17769:
17766:
17764:
17763:Weak topology
17761:
17759:
17756:
17754:
17751:
17749:
17746:
17745:
17743:
17739:
17732:
17728:
17725:
17723:
17720:
17718:
17715:
17713:
17710:
17708:
17705:
17703:
17700:
17698:
17695:
17693:
17690:
17688:
17687:Index theorem
17685:
17683:
17680:
17678:
17675:
17673:
17670:
17669:
17667:
17663:
17657:
17654:
17652:
17649:
17648:
17646:
17644:Open problems
17642:
17636:
17633:
17631:
17628:
17626:
17623:
17621:
17618:
17616:
17613:
17611:
17608:
17607:
17605:
17601:
17595:
17592:
17590:
17587:
17585:
17582:
17580:
17577:
17575:
17572:
17570:
17567:
17565:
17562:
17560:
17557:
17555:
17552:
17550:
17547:
17546:
17544:
17540:
17534:
17531:
17529:
17526:
17524:
17521:
17519:
17516:
17514:
17511:
17509:
17506:
17504:
17501:
17499:
17496:
17494:
17491:
17490:
17488:
17486:
17482:
17472:
17469:
17467:
17464:
17462:
17459:
17456:
17452:
17448:
17445:
17443:
17440:
17438:
17435:
17434:
17432:
17428:
17422:
17419:
17417:
17414:
17412:
17409:
17407:
17404:
17402:
17399:
17397:
17394:
17392:
17389:
17387:
17384:
17382:
17379:
17377:
17374:
17373:
17370:
17367:
17363:
17358:
17354:
17350:
17343:
17338:
17336:
17331:
17329:
17324:
17323:
17320:
17308:
17300:
17299:
17296:
17290:
17287:
17285:
17284:Sparse matrix
17282:
17280:
17277:
17275:
17272:
17270:
17267:
17266:
17264:
17262:
17258:
17252:
17249:
17247:
17244:
17242:
17239:
17237:
17234:
17232:
17229:
17227:
17224:
17223:
17221:
17219:constructions
17218:
17214:
17208:
17207:Outermorphism
17205:
17203:
17200:
17198:
17195:
17193:
17190:
17188:
17185:
17183:
17180:
17178:
17175:
17173:
17170:
17168:
17167:Cross product
17165:
17163:
17160:
17159:
17157:
17155:
17151:
17145:
17142:
17140:
17137:
17135:
17134:Outer product
17132:
17130:
17127:
17125:
17122:
17120:
17117:
17115:
17114:Orthogonality
17112:
17111:
17109:
17107:
17103:
17097:
17094:
17092:
17091:Cramer's rule
17089:
17087:
17084:
17082:
17079:
17077:
17074:
17072:
17069:
17067:
17064:
17062:
17061:Decomposition
17059:
17057:
17054:
17053:
17051:
17049:
17045:
17040:
17030:
17027:
17025:
17022:
17020:
17017:
17015:
17012:
17010:
17007:
17005:
17002:
17000:
16997:
16995:
16992:
16990:
16987:
16985:
16982:
16980:
16977:
16975:
16972:
16970:
16967:
16965:
16962:
16960:
16957:
16955:
16952:
16950:
16947:
16945:
16942:
16940:
16937:
16936:
16934:
16930:
16924:
16921:
16919:
16916:
16915:
16912:
16908:
16901:
16896:
16894:
16889:
16887:
16882:
16881:
16878:
16869:
16868:
16863:
16860:
16855:
16854:
16844:
16840:
16836:
16830:
16826:
16822:
16818:
16814:
16810:
16806:
16800:
16796:
16792:
16788:
16784:
16780:
16776:
16772:
16768:
16763:
16759:
16755:
16751:
16745:
16741:
16737:
16736:
16731:
16730:Rudin, Walter
16727:
16723:
16721:9780070542259
16717:
16713:
16708:
16707:
16701:
16700:Rudin, Walter
16697:
16693:
16689:
16685:
16679:
16675:
16670:
16666:
16664:0-7167-0344-0
16660:
16656:
16652:
16648:
16644:
16640:
16635:
16633:0-8218-1646-2
16629:
16625:
16621:
16617:
16613:
16609:
16603:
16599:
16595:
16591:
16590:Tu, Loring W.
16587:
16584:
16580:
16576:
16572:
16568:
16562:
16558:
16554:
16550:
16546:
16542:
16536:
16532:
16528:
16524:
16520:
16516:
16514:0-387-90093-4
16510:
16506:
16502:
16498:
16494:
16490:
16486:
16482:
16478:
16476:3-540-64243-9
16472:
16468:
16464:
16460:
16456:
16450:
16446:
16442:
16438:
16434:
16433:
16421:
16416:
16409:
16408:Schaefer 1966
16404:
16397:
16396:Schaefer 1966
16392:
16385:
16384:Schaefer 1966
16380:
16373:
16372:Bourbaki 2003
16368:
16361:
16356:
16349:
16348:Schaefer 1966
16344:
16342:
16334:
16329:
16327:
16319:
16303:
16299:
16274:
16271:
16267:
16262:
16251:
16247:
16241:
16234:
16233:Halmos (1974)
16229:
16222:
16221:Halmos (1974)
16217:
16210:
16205:
16197:
16191:
16187:
16180:
16173:
16168:
16162:
16157:
16150:
16149:Halmos (1974)
16145:
16138:
16137:Halmos (1974)
16133:
16127:p. 101, §3.94
16126:
16121:
16114:
16109:
16107:
16100:p. 37, §2.1.3
16099:
16094:
16087:
16082:
16080:
16075:
16061:
16057:
16051:
16044:
16038:
16019:
15999:
15960:
15954:
15952:
15950:
15948:
15939:
15935:
15930:
15922:
15916:
15909:
15891:
15887:
15865:
15862:
15853:
15840:
15818:
15815:
15806:
15790:
15770:
15765:
15761:
15752:
15739:
15716:
15712:
15689:
15681:
15670:
15666:
15665:
15646:
15642:
15631:
15627:
15616:
15613:
15611:
15608:
15606:
15603:
15601:
15598:
15596:
15593:
15591:
15588:
15586:
15583:
15582:
15576:
15574:
15569:
15563:
15543:
15540:
15537:
15534:
15530:
15524:
15518:
15511:
15508:
15504:
15501:
15494:
15493:
15492:
15489:
15483:
15479:
15473:
15468:
15463:
15458:
15453:
15447:
15441:
15438:
15433:
15429:
15425:
15421:
15417:
15412:
15408:
15403:
15399:
15395:
15390:
15388:
15384:
15379:
15375:
15368:
15364:
15358:
15354:
15335:
15331:
15328:
15324:
15321:
15315:
15312:
15309:
15306:
15302:
15296:
15290:
15287:
15281:
15272:
15259:
15258:
15257:
15256:, defined by
15254:
15249:
15244:
15240:
15230:
15217:
15208:
15192:
15186:
15183:
15175:
15171:
15167:
15162:
15158:
15151:
15148:
15142:
15128:
15124:
15113:
15110:
15101:
15097:
15074:
15067:
15059:
15052:
15048:
15039:
15032:
15026:
15021:
15012:
15009:
15003:
14994:
14980:
14977:
14973:
14969:
14963:
14958:
14937:
14933:
14929:
14922:
14919:
14912:
14909:
14902:
14901:
14900:
14898:
14893:
14888:
14884:
14880:
14876:
14871:
14864:
14860:
14853:
14846:
14842:
14837:
14833:
14829:
14824:
14820:
14815:
14811:
14805:
14801:
14781:
14775:
14772:
14769:
14766:
14763:
14760:
14756:
14753:
14749:
14746:
14740:
14735:
14731:
14723:
14722:
14721:
14718:
14713:
14709:
14705:
14695:
14691:
14687:
14683:
14676:
14672:
14668:
14661:
14657:
14653:
14646:
14642:
14638:
14632:
14627:
14621:
14617:
14613:
14608:
14602:
14598:
14594:
14590:
14585:
14581:, both duals
14579:
14575:
14569:
14565:
14559:
14553:
14547:
14542:
14538:
14534:
14515:
14510:
14506:
14502:
14498:
14495:
14491:
14486:
14482:
14478:
14473:
14469:
14461:
14460:
14459:
14457:
14453:
14449:
14444:
14439:
14435:
14428:
14423:
14421:
14416:
14411:
14407:
14403:
14399:
14394:
14391:
14386:
14382:
14376:
14372:
14368:
14363:
14357:
14353:
14349:
14344:
14340:
14321:
14317:
14314:
14310:
14306:
14303:
14299:
14295:
14288:
14285:
14282:
14272:
14271:
14270:
14268:
14264:
14259:
14253:
14248:
14244:
14239:
14233:
14226:
14222:
14202:
14198:
14195:
14191:
14188:
14184:
14181:
14178:
14175:
14172:
14166:
14159:
14156:
14148:
14147:
14146:
14143:
14139:
14133:
14129:
14122:
14118:
14108:
14106:
14101:
14099:
14095:
14091:
14087:
14082:
14079:
14071:
14067:
14066:supremum norm
14063:
14059:
14054:
14048:
14042:
14040:
14035:
14031:
14024:
14002:
13998:
13992:
13988:
13982:
13978:
13974:
13960:
13953:
13952:
13951:
13948:
13943:
13930:
13920:
13915:
13911:
13907:
13898:
13883:
13847:
13834:
13822:
13804:
13795:
13791:
13784:
13781:
13773:
13769:
13763:
13759:
13753:
13734:
13728:
13722:
13719:
13713:
13707:
13695:
13691:
13675:
13672:
13669:
13665:
13660:
13655:
13650:
13630:
13629:
13628:
13623:
13619:
13615:
13611:
13607:
13606:
13601:
13588:
13584:
13583:weak topology
13567:
13564:
13555:
13552:
13548:
13531:
13528:
13519:
13516:
13512:
13495:
13492:
13483:
13482:
13481:
13479:
13462:
13459:
13434:
13390:
13369:
13366:
13357:
13356:weak topology
13353:
13338:
13294:
13286:
13269:
13266:
13257:
13253:
13252:
13235:
13224:
13218:
13208:
13205:
13199:
13188:
13182:
13172:
13171:
13170:
13169:
13168:
13153:
13150:
13141:
13140:Hilbert space
13137:
13133:
13117:
13093:
13049:
13041:
13024:
13021:
13012:
13008:
13007:
13006:
13003:
12979:
12976:
12967:
12963:
12959:
12956:
12951:
12943:
12933:
12926:
12923:
12919:
12916:
12912:
12905:
12897:
12893:
12885:
12884:
12883:
12868:
12865:
12839:
12829:
12826:
12823:
12820:
12804:
12801:
12785:
12782:
12770:
12769:
12744:
12728:
12725:
12722:
12719:
12716:
12713:
12697:
12694:
12678:
12675:
12672:
12669:
12657:
12656:
12635:
12632:
12605:
12602:
12575:
12572:
12564:
12548:
12545:
12542:
12539:
12523:
12520:
12511:
12508:
12505:
12493:
12492:
12471:
12468:
12448:
12428:
12420:
12419:
12418:
12377:
12365:
12348:
12342:
12339:
12333:
12325:
12321:
12310:
12307:
12304:
12296:
12291:
12283:
12280:
12275:
12271:
12256:
12253:
12241:
12240:
12239:
12224:
12221:
12200:
12178:
12174:
12164:
12151:
12124:
12104:
12084:
12061:
12050:
12044:
12034:
12031:
12028:
12020:
12015:
12007:
11997:
11996:
11995:
11993:
11977:
11950:
11930:
11922:
11882:
11861:
11858:
11847:
11843:
11833:
11827:
11819:
11816:
11802:
11800:
11796:
11793:the space of
11780:
11776:
11760:
11744:
11717:
11713:
11698:and its dual
11673:
11672:distributions
11657:
11653:
11638:and its dual
11614:
11609:
11585:
11582:
11579:
11558:
11555:
11546:
11542:
11538:
11522:
11515:
11510:
11508:
11504:
11500:
11498:
11492:
11475:
11472:
11464:, denoted by
11449:
11445:
11397:
11384:
11380:
11370:
11368:
11352:
11349:
11344:
11341:
11337:
11333:
11330:
11327:
11324:
11316:
11312:
11294:
11290:
11286:
11276:
11272:
11254:
11245:
11241:
11237:
11233:
11232:dimensionless
11229:
11224:
11222:
11204:
11177:
11174:
11145:
11141:
11116:
11092:
11068:
11065:
11041:
11037:
11033:
11030:
11022:
11006:
11003:
10997:
10991:
10988:
10982:
10979:
10976:
10951:
10947:
10943:
10940:
10920:
10917:
10914:
10900:
10884:
10880:
10857:
10853:
10830:
10826:
10805:
10785:
10777:
10761:
10738:
10733:
10729:
10725:
10720:
10712:
10708:
10704:
10694:
10693:
10692:
10678:
10670:
10654:
10634:
10630:
10626:
10606:
10600:
10597:
10594:
10586:
10570:
10566:
10562:
10555:
10539:
10519:
10510:
10508:
10490:
10487:
10483:
10479:
10476:
10456:
10433:
10430:
10425:
10421:
10413:
10412:
10411:
10409:
10393:
10373:
10350:
10345:
10341:
10337:
10332:
10328:
10324:
10319:
10311:
10308:
10305:
10295:
10294:
10293:
10274:
10270:
10266:
10261:
10257:
10253:
10248:
10240:
10237:
10234:
10224:
10223:
10222:
10208:
10188:
10168:
10145:
10140:
10135:
10131:
10125:
10122:
10119:
10115:
10111:
10106:
10101:
10095:
10091:
10085:
10082:
10079:
10075:
10070:
10061:
10060:
10059:
10045:
10025:
10005:
9983:
9980:
9977:
9967:
9963:
9936:
9931:
9923:
9920:
9917:
9911:
9906:
9902:
9898:
9893:
9889:
9881:
9880:
9879:
9865:
9845:
9825:
9802:
9797:
9793:
9789:
9784:
9780:
9776:
9771:
9767:
9763:
9757:
9747:
9746:
9745:
9731:
9728:
9725:
9722:
9719:
9716:
9710:
9687:
9665:
9661:
9657:
9651:
9645:
9640:
9636:
9613:
9609:
9605:
9600:
9592:
9580:
9578:
9562:
9559:
9554:
9544:
9524:
9504:
9498:
9495:
9492:
9470:
9466:
9445:
9442:
9439:
9419:
9416:
9410:
9407:
9404:
9379:
9375:
9371:
9368:
9346:
9342:
9319:
9315:
9294:
9286:
9285:
9268:
9248:
9234:
9232:
9228:
9224:
9220:
9216:
9212:
9208:
9204:
9200:
9196:
9192:
9188:
9185:
9181:
9176:
9174:
9170:
9166:
9162:
9158:
9153:
9150:
9146:
9140:
9136:
9132:
9127:
9123:
9118:
9114:
9110:
9106:
9102:
9098:
9094:
9089:
9085:
9079:
9077:
9073:
9069:
9050:
9041:
9035:
9031:
9028:
9022:
9016:
9012:
9006:
8998:
8994:
8983:
8982:
8981:
8979:
8966:
8963:
8960:
8951:
8936:
8932:
8928:
8925:
8915:
8913:
8900:
8891:
8878:
8869:
8868:
8863:
8848:
8844:
8834:
8818:
8810:
8806:
8796:
8781:
8777:
8773:
8770:
8745:
8742:
8739:
8736:
8730:
8722:
8718:
8710:
8709:
8708:
8705:
8701:
8697:
8692:
8688:
8687:
8682:
8677:
8673:
8669:
8662:
8652:
8650:
8634:
8626:
8622:
8584:
8568:
8546:
8543:
8539:
8532:
8529:
8503:
8497:
8491:
8471:
8463:
8459:
8455:
8450:
8418:
8414:
8410:
8407:
8404:
8401:
8398:
8395:
8372:
8366:
8363:
8357:
8345:
8329:, defined by
8287:
8284:
8276:
8272:
8268:
8259:
8254:
8251:
8247:
8226:
8199:
8196:
8172:
8166:
8160:
8152:
8147:
8141:
8138:
8132:
8125:
8124:
8123:
8120:
8116:
8112:
8088:
8084:
8058:
8048:
8044:
8035:
8032:
8024:
8021:
8018:
8003:
8002:
8001:
7999:
7995:
7987:
7983:
7979:
7974:
7972:
7968:
7964:
7944:
7938:
7935:
7929:
7917:
7911:
7899:
7887:
7874:
7871:
7868:
7858:
7857:
7856:
7854:
7827:
7824:
7821:
7809:
7805:
7801:
7797:
7773:
7769:
7765:
7761:
7760:nondegenerate
7742:
7736:
7733:
7730:
7724:
7720:
7716:
7713:
7707:
7696:
7693:
7690:
7678:
7670:
7669:
7668:
7649:
7645:
7638:
7635:
7627:
7624:
7621:
7606:
7605:
7604:
7600:
7596:
7589:
7585:
7580:
7558:
7555:
7552:
7543:
7536:
7535:
7534:
7532:
7528:
7521:
7520:bilinear form
7517:
7513:
7509:
7505:
7495:
7493:
7489:
7473:
7460:
7456:
7433:
7425:
7400:
7392:
7388:
7370:
7364:
7342:
7323:
7312:
7304:
7291:
7287:
7250:
7240:
7236:
7227:
7217:
7202:
7194:
7186:
7178:
7172:
7151:
7150:
7149:
7147:
7131:
7111:
7102:
7099:
7097:
7093:
7089:
7084:
7082:
7078:
7057:
7053:
7049:
7046:
7041:
7038:
7035:
7031:
7027:
7022:
7018:
7012:
7009:
7006:
7002:
6998:
6993:
6988:
6984:
6979:
6976:
6973:
6969:
6964:
6959:
6954:
6950:
6942:
6941:
6940:
6926:
6906:
6898:
6880:
6876:
6852:
6849:
6844:
6841:
6838:
6834:
6830:
6825:
6815:
6811:
6804:
6801:
6794:
6793:
6792:
6778:
6758:
6750:
6732:
6722:
6718:
6705:
6691:
6669:
6665:
6641:
6636:
6632:
6626:
6622:
6616:
6613:
6610:
6606:
6602:
6594:
6590:
6583:
6578:
6574:
6568:
6565:
6562:
6558:
6554:
6550:
6544:
6532:
6528:
6522:
6519:
6516:
6512:
6507:
6503:
6496:
6495:
6494:
6480:
6460:
6438:
6434:
6430:
6424:
6418:
6398:
6392:
6389:
6386:
6366:
6341:
6328:
6325:
6320:
6316:
6295:
6275:
6255:
6235:
6227:
6209:
6205:
6184:
6175:
6161:
6158:
6155:
6135:
6115:
6090:
6078:
6074:
6068:
6065:
6062:
6058:
6050:
6049:
6048:
6034:
6014:
6011:
6008:
5985:
5979:
5976:
5971:
5967:
5946:
5940:
5937:
5934:
5927:of functions
5912:
5902:
5898:
5874:
5851:
5848:
5845:
5842:
5837:
5807:
5787:
5778:
5745:
5712:
5694:
5690:
5666:
5661:
5657:
5651:
5647:
5641:
5637:
5629:
5628:
5627:
5579:
5575:
5546:
5542:
5530:
5454:
5432:
5400:
5397:
5367:
5333:
5319:
5316:
5313:
5291:
5276:
5260:
5238:
5224:
5208:
5194:
5180:
5158:
5154:
5145:
5129:
5106:
5102:
5081:
5059:
5055:
5046:
5042:
5026:
5017:
5003:
4983:
4980:
4977:
4957:
4937:
4934:
4931:
4928:
4908:
4905:
4902:
4882:
4862:
4859:
4856:
4836:
4816:
4796:
4776:
4756:
4748:
4730:
4703:
4695:
4691:
4676:
4654:
4638:
4636:
4635:
4615:
4612:
4609:
4583:
4578:
4563:
4553:
4540:
4536:
4532:
4527:
4512:
4502:
4489:
4485:
4481:
4469:
4468:
4467:
4453:
4450:
4425:
4417:
4399:
4395:
4371:
4366:
4362:
4358:
4355:
4352:
4335:
4324:
4323:
4322:
4303:
4288:
4278:
4265:
4256:
4228:
4213:
4203:
4190:
4187:
4165:
4138:
4129:
4115:
4112:
4109:
4106:
4103:
4097:
4094:
4091:
4083:
4056:
4053:
4050:
4044:
4041:
4038:
4030:
4001:
3972:
3939:
3936:
3933:
3930:
3924:
3919:
3909:
3903:
3900:
3897:
3891:
3886:
3853:
3848:
3842:
3837:
3830:
3825:
3819:
3814:
3809:
3801:
3797:
3789:
3785:
3775:
3771:
3763:
3759:
3752:
3745:
3737:
3733:
3725:
3721:
3711:
3707:
3699:
3695:
3688:
3679:
3678:
3677:
3663:
3660:
3652:
3637:
3610:
3607:
3599:
3584:
3557:
3554:
3546:
3531:
3504:
3501:
3493:
3478:
3463:
3445:
3416:
3383:
3380:
3377:
3371:
3366:
3356:
3350:
3346:
3342:
3339:
3336:
3332:
3328:
3322:
3317:
3285:
3258:
3244:
3228:
3224:
3201:
3197:
3171:
3161:
3158:
3155:
3150:
3112:
3099:
3093:
3085:
3075:
3072:
3069:
3061:
3048:
3042:
3034:
3024:
3016:
3003:
2998:
2994:
2990:
2987:
2984:
2976:
2963:
2958:
2954:
2950:
2942:
2930:
2926:
2922:
2919:
2916:
2911:
2899:
2895:
2888:
2885:
2879:
2873:
2866:
2865:
2847:
2843:
2839:
2836:
2828:
2811:
2807:
2781:
2771:
2768:
2765:
2760:
2728:
2724:
2701:
2674:
2671:
2666:
2662:
2658:
2655:
2652:
2647:
2643:
2639:
2634:
2630:
2609:
2587:
2583:
2579:
2576:
2573:
2568:
2556:
2552:
2548:
2545:
2542:
2537:
2525:
2521:
2512:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2451:
2447:
2443:
2438:
2424:. Therefore,
2408:
2400:
2390:
2387:
2381:
2373:
2363:
2358:
2354:
2350:
2347:
2342:
2338:
2334:
2328:
2325:
2322:
2319:
2311:
2282:
2267:
2263:
2259:
2256:
2251:
2247:
2240:
2237:
2234:
2229:
2214:
2210:
2206:
2203:
2198:
2194:
2187:
2184:
2181:
2178:
2175:
2168:. Then also,
2153:
2149:
2145:
2139:
2131:
2102:
2098:
2094:
2088:
2080:
2051:
2039:
2035:
2031:
2028:
2025:
2020:
2008:
2004:
2000:
1997:
1975:
1963:
1959:
1955:
1952:
1949:
1944:
1932:
1928:
1924:
1921:
1901:
1898:
1895:
1892:
1889:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1834:
1819:
1818:
1817:
1801:
1797:
1787:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1751:
1746:
1742:
1738:
1730:
1718:
1714:
1710:
1707:
1704:
1699:
1687:
1683:
1674:
1659:
1640:
1630:
1627:
1624:
1619:
1584:
1574:
1571:
1568:
1563:
1544:
1543:
1537:
1536:
1532:
1530:
1526:
1508:
1503:
1499:
1473:
1468:
1464:
1460:
1452:
1437:
1423:
1422:
1421:
1407:
1404:
1399:
1395:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1349:
1344:
1340:
1336:
1328:
1316:
1312:
1308:
1305:
1302:
1297:
1285:
1281:
1272:
1258:
1257:
1256:
1242:
1217:
1207:
1204:
1201:
1196:
1178:
1175:, called the
1160:
1156:
1135:
1110:
1100:
1097:
1094:
1089:
1072:
1056:
1034:
1030:
1009:
999:
989:
987:
971:
963:
959:
955:
952:
949:
943:
940:
937:
927:
923:
907:
904:
901:
895:
889:
883:
874:
858:
855:
852:
846:
840:
834:
825:
812:
803:
790:
767:
763:
754:
741:
731:
729:
728:
723:
719:
701:
697:
687:
673:
670:
667:
659:
646:
643:
640:
617:
613:
609:
606:
603:
600:
572:
565:
559:
555:
551:
548:
546:
538:
529:
526:
513:
507:
504:
498:
492:
489:
487:
479:
470:
467:
464:
450:
449:
448:
446:
433:
410:
406:
382:
379:
376:
370:
367:
359:
358:homomorphisms
355:
351:
338:
332:
329:
326:
318:
301:
298:
275:
271:
248:
244:
236:
220:
213:
197:
190:
180:
178:
174:
170:
166:
165:adjoint space
162:
158:
154:
149:
147:
143:
139:
138:distributions
135:
131:
127:
122:
120:
117:, called the
116:
112:
106:
103:
101:
97:
84:
81:
72:
68:
64:
60:
47:
39:
35:
30:
19:
17987:Mackey–Arens
17975:Main results
17851:
17753:Balanced set
17727:Distribution
17665:Applications
17518:Krein–Milman
17503:Closed graph
17446:
17225:
17217:Vector space
16949:Vector space
16865:
16824:
16790:
16778:
16766:
16734:
16705:
16673:
16654:
16623:
16593:
16552:
16526:
16500:
16488:
16466:
16440:
16430:Bibliography
16415:
16403:
16391:
16379:
16367:
16355:
16253:
16246:Tao, Terence
16240:
16228:
16216:
16204:
16185:
16179:
16167:
16156:
16144:
16132:
16125:Axler (2015)
16120:
16093:
16055:
16050:
16037:
15937:
15933:
15915:
15833:
15805:Halmos (1974
15732:
15662:
15630:
15567:
15561:
15558:
15487:
15481:
15477:
15471:
15466:
15461:
15456:
15451:
15445:
15442:
15436:
15431:
15423:
15415:
15410:
15406:
15401:
15393:
15391:
15377:
15373:
15366:
15362:
15350:
15252:
15247:
15242:
15238:
15234:
15210:
15080:
15072:
15065:
15057:
15050:
15047:ordered pair
15045:denotes the
15037:
15030:
15007:
15001:
14986:
14975:
14971:
14967:
14961:
14954:
14896:
14891:
14886:
14882:
14878:
14874:
14869:
14862:
14858:
14851:
14844:
14840:
14835:
14827:
14822:
14818:
14813:
14809:
14803:
14799:
14796:
14716:
14711:
14707:
14703:
14702:Assume that
14701:
14698:Annihilators
14689:
14685:
14681:
14674:
14670:
14666:
14659:
14655:
14651:
14644:
14640:
14636:
14630:
14625:
14619:
14615:
14611:
14606:
14600:
14596:
14592:
14583:
14577:
14573:
14567:
14563:
14557:
14551:
14545:
14540:
14536:
14532:
14530:
14451:
14447:
14442:
14437:
14430:
14426:
14424:
14414:
14409:
14405:
14397:
14395:
14389:
14384:
14374:
14370:
14366:
14361:
14355:
14351:
14347:
14342:
14338:
14336:
14266:
14262:
14257:
14251:
14246:
14242:
14237:
14231:
14224:
14220:
14217:
14141:
14137:
14131:
14127:
14124:
14102:
14083:
14077:
14069:
14057:
14052:
14046:
14043:
14033:
14026:
14022:
14019:
13946:
13923:
13918:
13913:
13909:
13905:
13869:
13820:
13771:
13767:
13761:
13757:
13751:
13749:
13621:
13617:
13613:
13603:
13599:
13597:
13550:
13514:
13450:
13136:Banach space
13109:
13004:
13001:
12856:
12392:
12165:
12076:
11994:of the form
11849:
11808:
11610:
11544:
11540:
11536:
11511:
11496:
11490:
11376:
11274:
11244:unit of time
11225:
10906:
10753:
10511:
10448:
10365:
10291:
10160:
9951:
9817:
9581:
9282:
9240:
9230:
9226:
9222:
9218:
9214:
9210:
9206:
9198:
9194:
9190:
9186:
9179:
9177:
9172:
9168:
9164:
9151:
9148:
9144:
9125:
9121:
9116:
9108:
9104:
9100:
9096:
9091:produces an
9087:
9083:
9080:
9071:
9067:
9065:
8953:
8918:
8916:
8893:
8871:
8865:
8836:
8798:
8763:
8761:
8703:
8699:
8695:
8690:
8684:
8675:
8671:
8667:
8664:
8599:. This map
8198:homomorphism
8192:
8118:
8114:
8110:
8073:
7993:
7980:is over the
7977:
7975:
7970:
7966:
7962:
7960:
7852:
7807:
7803:
7799:
7795:
7771:
7767:
7763:
7757:
7666:
7598:
7594:
7587:
7583:
7581:taking each
7578:
7576:
7530:
7526:
7511:
7507:
7503:
7501:
7338:
7103:
7100:
7087:
7085:
7074:
6867:
6706:
6656:
6225:
6176:
6107:
5779:
5681:
5528:
5334:
5200:
5045:level curves
5018:
4875:matrix, and
4693:
4690:real numbers
4639:
4632:
4598:
4386:
4130:
3868:
3250:
3132:
1788:
1660:
1545:
1528:
1490:
1386:
1001:
876:
827:
805:
783:
734:
732:
727:linear forms
725:
721:
717:
688:
633:
592:
426:
319:
234:
189:vector space
186:
172:
168:
164:
160:
157:polarer Raum
156:
152:
150:
123:
118:
104:
74:
71:linear forms
66:
62:
40:
38:vector space
31:
29:
17966:Ultrastrong
17949:Strong dual
17857:Dual system
17682:Heat kernel
17672:Hardy space
17579:Trace class
17493:Hahn–Banach
17455:Topological
17197:Multivector
17162:Determinant
17119:Dot product
16964:Linear span
16655:Gravitation
16549:Lang, Serge
16422:, chapter 4
16115:p. 19, §3.1
16043:functionals
15931:defined on
15908:Trèves 2006
15590:Dual module
15573:reflexivity
15385:are called
15015:Double dual
13598:Let 1 <
12421:Each point
11846:Dual system
10018:indexed by
9458:. That is,
9284:annihilator
9113:isomorphism
8683:, then the
8625:isomorphism
8193:There is a
7667:defined by
6919:indexed by
5867:identifies
2716:results in
2066:to scalars
984:called the
317:linear maps
34:mathematics
18055:Categories
17897:Topologies
17852:Dual space
17615:C*-algebra
17430:Properties
17231:Direct sum
17066:Invertible
16969:Linear map
16583:0984.00001
16420:Rudin 1991
16360:Rudin 1973
16223:pp. 25, 28
16195:0201006391
16151:p. 21, §14
16139:p. 20, §13
16070:References
15753:, so that
15475:for every
15359:, meaning
15023:This is a
14832:surjection
14664:, or from
14115:See also:
14062:convergent
13627:for which
13587:dual pairs
13551:stereotype
11826:completion
11805:Properties
11545:dual space
11543:, or just
11495:Euclidean
11489:. For any
11383:continuous
10776:direct sum
10667:is in the
9537:vanishes:
9394:such that
8762:for every
8681:linear map
8619:is always
8122:such that
7096:isomorphic
6749:direct sum
6148:, and any
5959:such that
5746:, whereas
5144:level sets
4970:must be a
3958:. Because
3189:generates
1177:dual basis
1069:. Given a
998:Dual basis
996:See also:
187:Given any
175:is due to
67:dual space
18024:Total set
17910:Dual norm
17877:Polar set
17589:Unbounded
17584:Transpose
17542:Operators
17471:Separable
17466:Reflexive
17451:Algebraic
17437:Barrelled
17261:Numerical
17024:Transpose
16867:MathWorld
16843:853623322
16823:(2006) .
16813:840278135
16692:144216834
16499:(1974) .
16272:−
16113:Tu (2011)
15892:∗
15791:ω
15771:ω
15766:∗
15717:∗
15690:∗
15682:⋅
15647:∨
15595:Dual norm
15538:∈
15519:φ
15516:↦
15505:∈
15502:φ
15428:Hausdorff
15387:reflexive
15383:bijection
15325:∈
15322:φ
15310:∈
15291:φ
15282:φ
15267:Ψ
15237:Ψ :
15218:φ
15187:φ
15152:φ
15143:φ
15118:Ψ
15091:Ψ
14993:separable
14938:⊥
14913:
14865: )′
14776:φ
14773:
14767:⊆
14750:∈
14747:φ
14736:⊥
14587:have the
14503:∘
14487:∗
14479:∘
14311:∘
14286:∘
14192:∈
14189:φ
14179:∘
14176:φ
14167:φ
13979:∑
13961:φ
13931:φ
13835:φ
13792:ℓ
13785:∈
13782:φ
13732:∞
13681:∞
13666:∑
13647:‖
13638:‖
13610:sequences
13515:reflexive
13403:(so here
13307:(so here
13219:φ
13206:≤
13203:‖
13197:‖
13186:‖
13183:φ
13180:‖
13062:(so here
12980:∈
12948:‖
12944:φ
12941:‖
12920:∈
12917:φ
12830:∈
12824:⋅
12821:λ
12805:∈
12802:λ
12786:∈
12723:⊆
12717:∪
12698:∈
12679:∈
12636:∈
12606:∈
12576:∈
12543:∈
12524:∈
12509:∈
12472:∈
12372:∞
12369:→
12362:⟶
12343:φ
12340:−
12322:φ
12308:∈
12288:‖
12284:φ
12281:−
12272:φ
12268:‖
12257:∈
12201:φ
12175:φ
12085:φ
12045:φ
12032:∈
12012:‖
12008:φ
12005:‖
11992:seminorms
11815:Hausdorff
11589:→
11580:φ
11450:∗
11342:−
11331:⋅
11271:frequency
11175:−
11146:∗
11042:∗
11034:⊕
11004:∈
10992:φ
10986:⟩
10983:φ
10974:⟨
10952:∗
10944:∈
10941:φ
10918:∈
10831:∗
10726:≅
10721:∗
10604:→
10552:then the
10491:∗
10488:∗
10480:≈
10309:∩
10267:∩
10123:∈
10116:⋂
10083:∈
10076:⋃
9981:∈
9921:∩
9912:⊆
9798:∗
9790:⊆
9777:⊆
9764:⊆
9729:⊆
9723:⊆
9717:⊆
9666:∗
9658:⊆
9614:∗
9502:→
9443:∈
9380:∗
9372:∈
9320:∗
9203:transpose
9093:injective
9029:φ
9007:φ
8999:∗
8964:∈
8937:∗
8929:∈
8926:φ
8879:φ
8849:∗
8819:φ
8811:∗
8782:∗
8774:∈
8771:φ
8743:∘
8740:φ
8731:φ
8723:∗
8686:transpose
8621:injective
8607:Ψ
8572:↦
8547:∗
8544:∗
8536:→
8527:Ψ
8498:φ
8495:↦
8492:φ
8469:→
8464:∗
8419:∗
8411:∈
8408:φ
8399:∈
8367:φ
8358:φ
8340:Ψ
8291:Φ
8282:→
8277:∗
8266:Φ
8255:∗
8252:∗
8207:Ψ
8156:¯
8153:α
8139:α
8094:¯
8089:∗
8054:¯
8049:∗
8039:→
8028:⟩
8025:⋅
8019:⋅
8016:⟨
8012:Φ
7996:with the
7924:Φ
7894:Φ
7883:Φ
7879:⟩
7866:⟨
7836:Φ
7832:⟩
7828:⋅
7822:⋅
7819:⟨
7782:Φ
7740:⟩
7728:⟨
7700:⟩
7697:⋅
7691:⋅
7688:⟨
7684:Φ
7650:∗
7642:→
7631:⟩
7628:⋅
7622:⋅
7619:⟨
7615:Φ
7562:⟩
7559:⋅
7550:⟨
7547:↦
7461:∗
7434:≤
7393:∗
7292:∗
7241:∗
7050:≅
7039:∈
7036:α
7032:∏
7028:≅
7023:∗
7010:∈
7007:α
7003:∏
6999:≅
6994:∗
6977:∈
6974:α
6970:⨁
6960:≅
6955:∗
6842:∈
6839:α
6835:⨁
6831:≅
6805:≅
6692:α
6670:α
6637:α
6633:θ
6627:α
6614:∈
6611:α
6607:∑
6595:α
6579:α
6566:∈
6563:α
6559:∑
6545:α
6533:α
6520:∈
6517:α
6513:∑
6439:α
6435:θ
6425:α
6419:θ
6396:→
6387:θ
6342:α
6321:α
6317:θ
6159:∈
6091:α
6079:α
6066:∈
6063:α
6059:∑
6012:∈
6009:α
5986:α
5972:α
5944:→
5849:∈
5846:α
5838:α
5728:∞
5711:dimension
5638:∑
5612:∞
5482:∞
5401:∈
5366:sequences
5350:∞
5317:∈
5314:α
5292:α
5277:elements
5239:α
5159:∗
5107:∗
5060:∗
4981:×
4906:×
4860:×
4619:⟩
4616:⋅
4610:⋅
4607:⟨
4569:⟩
4546:⟨
4537:∑
4518:⟩
4495:⟨
4486:∑
4451:∈
4418:of order
4353:⋅
4339:^
4289:⋯
4260:^
4214:⋯
4107:−
3931:−
3462:one-forms
3229:∗
3202:∗
3159:…
3073:⋯
2995:α
2988:⋯
2955:α
2927:α
2920:⋯
2896:α
2848:∗
2840:∈
2812:∗
2769:…
2725:λ
2663:λ
2656:⋯
2644:λ
2631:λ
2588:∗
2580:∈
2553:λ
2546:⋯
2522:λ
2492:…
2452:∗
2444:∈
2391:λ
2355:β
2351:λ
2339:α
2326:λ
2264:β
2260:λ
2248:α
2238:⋯
2211:β
2207:λ
2195:α
2182:λ
2150:β
2099:α
2036:β
2029:⋯
2005:β
1960:α
1953:⋯
1929:α
1899:∈
1861:…
1816:because:
1802:∗
1768:…
1708:⋯
1628:…
1572:…
1546:Consider
1500:δ
1465:δ
1405:∈
1366:…
1306:⋯
1205:…
1161:∗
1098:…
1035:∗
969:→
964:∗
956:×
947:⟩
944:⋅
938:⋅
935:⟨
911:⟩
908:φ
899:⟨
884:φ
859:φ
835:φ
768:∗
742:φ
722:one-forms
718:covectors
702:∗
671:∈
644:∈
618:∗
610:∈
607:ψ
601:φ
560:φ
530:φ
508:ψ
493:φ
471:ψ
465:φ
411:∗
371:
336:→
327:φ
276:∨
249:∗
100:pointwise
65:(or just
17959:operator
17932:operator
17791:Category
17603:Algebras
17485:Theorems
17442:Complete
17411:Schwartz
17357:glossary
17307:Category
17246:Subspace
17241:Quotient
17192:Bivector
17106:Bilinear
17048:Matrices
16923:Glossary
16777:(1966).
16758:21163277
16732:(1991).
16702:(1973).
16653:(1973).
16622:(1999).
16598:Springer
16592:(2011).
16551:(2002),
16505:Springer
16487:(2003).
16465:(1989).
16445:Springer
16439:(2015).
16410:, IV.1.2
16386:, IV.5.5
15866:′
15819:′
15579:See also
15512:′
15371:for all
15365:) ‖ = ‖
15357:isometry
15332:′
15209:for any
15114:′
15043:⟩
15029:⟨
15011:is not.
14923:′
14757:′
14499:′
14318:′
14307:′
14296:′
14199:′
14160:′
14140: :
14130: :
14020:for all
13805:′
13594:Examples
13568:′
13532:′
13496:′
13463:′
13370:′
13270:′
13154:′
13025:′
12927:′
12869:′
12225:′
11862:′
11777:′
11714:′
11654:′
11559:′
11476:′
9432:for all
8867:pullback
8698: :
8670: :
8388:for all
8113: :
7601:⟩
7593:⟨
6707:The set
4769:-vector
2513:Suppose
1914:such as
593:for all
302:′
177:Bourbaki
155:include
134:measures
18012:Subsets
17944:Mackey
17867:Duality
17833:Duality
17594:Unitary
17574:Nuclear
17559:Compact
17554:Bounded
17549:Adjoint
17523:Min–max
17416:Sobolev
17401:Nuclear
17391:Hilbert
17386:Fréchet
17351: (
16918:Outline
16624:Algebra
16575:1878556
16553:Algebra
16374:, II.42
16211:, §VI.4
15669:Tu 2011
15418:is not
15400:then Ψ(
14847:
14806:
14456:adjoint
14402:compact
14261:. When
14142:W′ → V′
14103:By the
14084:By the
13608:of all
11311:seconds
11307:
11279:
11267:
11247:
11217:
11197:
11193:
11164:
11160:
11133:
11129:
11109:
11105:
11085:
11081:
11058:
10818:, then
10410:, then
10058:, then
9744:, then
9205:matrix
9197:, then
9171:) with
9131:algebra
9076:adjoint
8519:, then
8195:natural
7982:complex
7081:modules
5041:vectors
4414:is the
1523:is the
210:over a
17837:linear
17569:Normal
17406:Orlicz
17396:Hölder
17376:Banach
17365:Spaces
17353:topics
17202:Tensor
17014:Kernel
16944:Vector
16939:Scalar
16841:
16831:
16811:
16801:
16756:
16746:
16718:
16690:
16680:
16661:
16630:
16604:
16581:
16573:
16563:
16537:
16511:
16473:
16451:
16398:, IV.1
16350:, II.4
16335:, II.2
16192:
16174:, §2.5
16060:kernel
15319:
14238:T → T′
14229:is in
14068:) and
13868:where
12937:
12931:
12909:
12903:
12117:, and
12077:where
11512:For a
11381:, the
10669:kernel
9281:. The
9184:matrix
9133:under
8892:along
8294:
7088:always
6411:(with
5709:. The
5390:. For
4849:as an
4829:, and
4387:where
3623:, and
2862:. Then
1491:where
660:, and
233:, the
179:1938.
167:, and
140:, and
126:tensor
36:, any
17927:polar
17381:Besov
17071:Minor
17056:Block
16994:Basis
16362:, 3.1
15622:Notes
15396:is a
15392:When
14959:that
14885:into
14877:. If
14867:into
14834:from
14531:When
14436:from
14425:When
14400:is a
14396:When
14337:When
14132:V → W
14037:(see
13138:or a
13130:is a
11813:is a
11539:, or
11499:space
11056:is a
10774:is a
10406:is a
10221:then
9878:then
8679:is a
8219:from
7794:from
7766:. If
5223:basis
4180:, if
1539:Proof
1071:basis
724:, or
212:field
17996:Maps
17922:Weak
17839:maps
17729:(or
17447:Dual
17226:Dual
17081:Rank
16839:OCLC
16829:ISBN
16809:OCLC
16799:ISBN
16754:OCLC
16744:ISBN
16716:ISBN
16688:OCLC
16678:ISBN
16659:ISBN
16628:ISBN
16602:ISBN
16561:ISBN
16535:ISBN
16509:ISBN
16471:ISBN
16449:ISBN
16190:ISBN
15634:For
15361:‖ Ψ(
15056:and
14571:and
14555:and
14539:and
14408:and
14341:and
14265:and
14119:and
14032:) ∈
13917:) ∈
13760:+ 1/
13729:<
13354:The
13254:The
13009:The
12957:<
12595:and
11844:and
11757:the
11535:its
11415:(or
10872:and
10798:and
10292:and
10181:and
9838:and
9241:Let
9213:and
9193:and
9147:) =
8952:and
8691:dual
8689:(or
7371:<
7195:<
4694:rows
4069:and
3987:and
3460:are
3431:and
3133:and
2466:for
2297:and
2117:and
1990:and
1820:The
173:dual
153:dual
16795:GTM
16579:Zbl
16318:...
16235:§44
16054:If
15568:V′′
15462:V′′
15459:to
15452:V′′
15426:is
15253:V′′
15243:V′′
15063:to
14991:is
14910:ker
14770:ker
14714:in
14679:to
14649:to
14450:on
14364:in
14255:to
14245:to
14125:If
14041:).
14025:= (
13944:on
13908:= (
13764:= 1
13754:by
13616:= (
13556:If
13520:If
13484:If
13358:on
13287:in
13258:on
13193:sup
13110:If
13042:in
13013:on
12441:of
12301:sup
12213:in
12025:sup
11923:of
11919:of
11828:of
11809:If
11275:per
11107:is
10671:of
10512:If
10366:If
9952:If
9818:If
9307:in
9287:of
9107:to
9099:to
8870:of
8835:in
8665:If
8627:if
7855:by
7851:on
7806:if
7591:to
7525:on
7502:If
7104:If
6493:by
6473:on
6288:on
6248:to
6226:all
6224:of
6108:in
5742:is
5713:of
5597:of
5529:all
5201:If
5019:If
4696:of
4151:is
3271:is
3243:.
1128:in
1002:If
875:or
804:of
368:hom
290:or
73:on
32:In
18057::
17355:–
16864:.
16837:.
16807:.
16793:.
16752:.
16742:.
16714:.
16686:.
16649:;
16645:;
16618:;
16600:.
16577:,
16571:MR
16569:,
16555:,
16533:.
16529:.
16507:.
16447:.
16340:^
16325:^
16252:.
16105:^
16078:^
15946:^
15936:×
15923:,
15803:.
15562:V′
15488:V′
15480:∈
15472:V′
15446:V′
15437:V′
15409:∈
15389:.
15376:∈
15241:→
15071:+
15036:,
14979:.
14976:W′
14974:→
14968:V′
14962:j′
14899::
14892:j′
14870:V′
14852:P′
14819:V′
14720:,
14717:V′
14694:.
14688:,
14686:V′
14673:,
14671:W′
14658:,
14656:V′
14643:,
14641:W′
14631:T′
14618:,
14616:X′
14599:,
14597:X′
14584:X′
14576:=
14566:=
14558:V′
14552:W′
14546:T′
14443:V′
14422:.
14415:T′
14390:T′
14373:,
14356:V′
14354:,
14352:W′
14258:V′
14252:W′
14232:V′
14221:T′
14138:T′
14100:.
14081:.
13756:1/
13447:).
13351:).
13106:).
12655::
12491::
12378:0.
11832:.
11801:.
11608:.
11497:n-
11369:.
11223:.
10989::=
10899:.
10426:00
9579:.
9145:fg
9124:=
9086:↦
9078:.
8980::
8914:.
8702:→
8693:)
8674:→
8117:→
7973:.
7597:,
7586:∈
7494:.
6704:.
5418:,
4895:a
4637:.
4128:.
3802:22
3790:12
3776:21
3764:11
3738:22
3726:21
3712:12
3700:11
3570:,
3517:,
1786:.
1531:.
988:.
730:.
720:,
686:.
632:,
163:,
159:,
148:.
136:,
121:.
17825:e
17818:t
17811:v
17733:)
17457:)
17453:/
17449:(
17359:)
17341:e
17334:t
17327:v
16899:e
16892:t
16885:v
16870:.
16845:.
16815:.
16760:.
16724:.
16694:.
16667:.
16638:.
16636:.
16610:.
16543:.
16517:.
16479:.
16457:.
16304:T
16300:V
16275:1
16268:T
16263:V
16198:.
16056:V
16020:V
16000:V
15977:N
15971:R
15941:.
15938:V
15934:V
15888:V
15863:V
15841:V
15816:V
15762:F
15740:F
15713:F
15686:)
15679:(
15667:(
15643:V
15544:,
15541:V
15535:x
15531:,
15528:)
15525:x
15522:(
15509:V
15482:V
15478:x
15467:x
15457:V
15432:V
15424:V
15416:V
15411:V
15407:x
15402:x
15394:V
15378:V
15374:x
15369:‖
15367:x
15363:x
15336:.
15329:V
15316:,
15313:V
15307:x
15303:,
15300:)
15297:x
15294:(
15288:=
15285:)
15279:(
15276:)
15273:x
15270:(
15248:V
15239:V
15196:)
15193:x
15190:(
15184:=
15181:)
15176:2
15172:x
15168:+
15163:1
15159:x
15155:(
15149:=
15146:)
15140:(
15137:]
15134:)
15129:2
15125:x
15121:(
15111:+
15107:)
15102:1
15098:x
15094:(
15088:[
15076:2
15073:x
15069:1
15066:x
15061:2
15058:x
15054:1
15051:x
15041:2
15038:x
15034:1
15031:x
15008:ℓ
15002:ℓ
14997:V
14989:V
14972:W
14934:W
14930:=
14927:)
14920:j
14916:(
14897:W
14887:V
14883:W
14879:j
14875:W
14863:W
14859:V
14857:(
14845:W
14841:V
14836:V
14828:P
14823:W
14814:W
14810:W
14804:W
14800:V
14782:.
14779:}
14764:W
14761::
14754:V
14744:{
14741:=
14732:W
14712:W
14708:V
14704:W
14692:)
14690:V
14684:(
14682:σ
14677:)
14675:W
14669:(
14667:σ
14662:)
14660:V
14654:(
14652:β
14647:)
14645:W
14639:(
14637:β
14626:X
14622:)
14620:X
14614:(
14612:σ
14607:X
14603:)
14601:X
14595:(
14593:β
14578:W
14574:X
14568:V
14564:X
14541:W
14537:V
14533:T
14516:.
14511:V
14507:i
14496:T
14492:=
14483:T
14474:V
14470:i
14452:V
14448:T
14438:V
14433:V
14431:i
14427:V
14410:W
14406:V
14398:T
14385:T
14377:)
14375:W
14371:V
14369:(
14367:L
14362:T
14358:)
14350:(
14348:L
14343:W
14339:V
14322:.
14315:U
14304:T
14300:=
14293:)
14289:T
14283:U
14280:(
14267:U
14263:T
14247:W
14243:V
14227:)
14225:φ
14223:(
14203:.
14196:W
14185:,
14182:T
14173:=
14170:)
14164:(
14157:T
14128:T
14078:ℓ
14073:0
14070:c
14058:c
14053:ℓ
14047:ℓ
14034:ℓ
14029:n
14027:b
14023:b
14003:n
13999:b
13993:n
13989:a
13983:n
13975:=
13972:)
13968:b
13964:(
13947:ℓ
13919:ℓ
13914:n
13910:a
13906:a
13901:n
13884:n
13879:e
13856:)
13853:)
13848:n
13843:e
13838:(
13832:(
13821:ℓ
13802:)
13796:p
13788:(
13772:ℓ
13768:ℓ
13762:q
13758:p
13752:q
13735:.
13723:p
13720:1
13714:)
13708:p
13703:|
13696:n
13692:a
13687:|
13676:0
13673:=
13670:n
13661:(
13656:=
13651:p
13642:a
13625:)
13622:n
13618:a
13614:a
13605:ℓ
13600:p
13565:V
13553:.
13529:V
13517:.
13493:V
13460:V
13435:V
13413:A
13391:V
13367:V
13339:V
13317:A
13295:V
13267:V
13236:.
13232:|
13228:)
13225:x
13222:(
13215:|
13209:1
13200:x
13189:=
13151:V
13118:V
13094:V
13072:A
13050:V
13022:V
12985:A
12977:A
12968:,
12964:}
12960:1
12952:A
12934::
12924:V
12913:{
12906:=
12898:A
12894:U
12866:V
12840:.
12835:A
12827:A
12810:F
12791:A
12783:A
12754:A
12729:.
12726:C
12720:B
12714:A
12703:A
12695:C
12684:A
12676:B
12673:,
12670:A
12641:A
12633:C
12611:A
12603:B
12581:A
12573:A
12549:.
12546:A
12540:x
12529:A
12521:A
12512:V
12506:x
12477:A
12469:A
12449:V
12429:x
12403:A
12366:i
12356:|
12352:)
12349:x
12346:(
12337:)
12334:x
12331:(
12326:i
12317:|
12311:A
12305:x
12297:=
12292:A
12276:i
12262:A
12254:A
12222:V
12179:i
12152:.
12147:A
12125:A
12105:V
12062:,
12058:|
12054:)
12051:x
12048:(
12041:|
12035:A
12029:x
12021:=
12016:A
11978:,
11973:A
11951:V
11931:V
11905:A
11883:V
11859:V
11830:X
11822:X
11811:X
11781:,
11772:S
11745:,
11740:S
11718:,
11709:E
11684:E
11658:,
11649:D
11624:D
11594:F
11586:V
11583::
11556:V
11523:V
11473:V
11446:V
11424:R
11402:C
11398:=
11394:F
11353:6
11350:=
11345:1
11338:s
11334:2
11328:s
11325:3
11295:t
11291:/
11287:1
11255:t
11205:V
11181:)
11178:n
11172:(
11142:V
11117:n
11093:V
11069:n
11066:2
11038:V
11031:V
11007:F
11001:)
10998:x
10995:(
10980:,
10977:x
10948:V
10921:V
10915:v
10885:0
10881:B
10858:0
10854:A
10827:V
10806:B
10786:A
10762:V
10739:.
10734:0
10730:W
10717:)
10713:W
10709:/
10705:V
10702:(
10679:f
10655:W
10635:W
10631:/
10627:V
10607:F
10601:V
10598::
10595:f
10571:W
10567:/
10563:V
10540:V
10520:W
10484:V
10477:V
10457:W
10434:W
10431:=
10422:W
10394:W
10374:V
10351:.
10346:0
10342:B
10338:+
10333:0
10329:A
10325:=
10320:0
10316:)
10312:B
10306:A
10303:(
10275:0
10271:B
10262:0
10258:A
10254:=
10249:0
10245:)
10241:B
10238:+
10235:A
10232:(
10209:V
10189:B
10169:A
10146:.
10141:0
10136:i
10132:A
10126:I
10120:i
10112:=
10107:0
10102:)
10096:i
10092:A
10086:I
10080:i
10071:(
10046:I
10026:i
10006:V
9984:I
9978:i
9974:)
9968:i
9964:A
9960:(
9937:.
9932:0
9928:)
9924:B
9918:A
9915:(
9907:0
9903:B
9899:+
9894:0
9890:A
9866:V
9846:B
9826:A
9803:.
9794:V
9785:0
9781:S
9772:0
9768:T
9761:}
9758:0
9755:{
9732:V
9726:T
9720:S
9714:}
9711:0
9708:{
9688:V
9662:V
9655:}
9652:0
9649:{
9646:=
9641:0
9637:V
9610:V
9606:=
9601:0
9597:}
9593:0
9590:{
9563:0
9560:=
9555:S
9550:|
9545:f
9525:S
9505:F
9499:V
9496::
9493:f
9471:0
9467:S
9446:S
9440:s
9420:0
9417:=
9414:]
9411:s
9408:,
9405:f
9402:[
9376:V
9369:f
9347:0
9343:S
9316:V
9295:S
9269:V
9249:S
9231:R
9227:f
9223:A
9219:f
9215:V
9211:W
9207:A
9199:f
9195:W
9191:V
9187:A
9180:f
9173:f
9169:f
9165:F
9152:f
9149:g
9143:(
9126:W
9122:V
9117:W
9109:V
9105:W
9101:W
9097:V
9088:f
9084:f
9072:W
9068:V
9051:,
9048:]
9045:)
9042:v
9039:(
9036:f
9032:,
9026:[
9023:=
9020:]
9017:v
9013:,
9010:)
9004:(
8995:f
8991:[
8967:V
8961:v
8933:W
8901:f
8845:V
8822:)
8816:(
8807:f
8778:W
8746:f
8737:=
8734:)
8728:(
8719:f
8704:V
8700:W
8696:f
8676:W
8672:V
8668:f
8635:V
8585:v
8580:v
8577:e
8569:v
8540:V
8533:V
8530::
8507:)
8504:v
8501:(
8472:F
8460:V
8456::
8451:v
8446:v
8443:e
8415:V
8405:,
8402:V
8396:v
8376:)
8373:v
8370:(
8364:=
8361:)
8355:(
8352:)
8349:)
8346:v
8343:(
8337:(
8317:}
8313:r
8310:a
8307:e
8304:n
8301:i
8298:l
8288::
8285:F
8273:V
8269::
8263:{
8260:=
8248:V
8227:V
8173:.
8170:)
8167:v
8164:(
8161:f
8148:=
8145:)
8142:v
8136:(
8133:f
8119:C
8115:V
8111:f
8085:V
8059:.
8045:V
8036:V
8033::
8022:,
7994:V
7978:V
7971:V
7967:V
7963:V
7945:.
7942:]
7939:w
7936:,
7933:)
7930:v
7927:(
7921:[
7918:=
7915:)
7912:w
7909:(
7906:)
7903:)
7900:v
7897:(
7891:(
7888:=
7875:w
7872:,
7869:v
7853:V
7825:,
7808:V
7804:V
7800:V
7796:V
7772:V
7768:V
7764:V
7743:.
7737:w
7734:,
7731:v
7725:=
7721:]
7717:w
7714:,
7711:)
7708:v
7705:(
7694:,
7679:[
7646:V
7639:V
7636::
7625:,
7599:w
7595:v
7588:V
7584:w
7579:V
7556:,
7553:v
7544:v
7531:V
7527:V
7512:V
7508:V
7504:V
7474:,
7470:|
7466:)
7457:V
7453:(
7449:m
7446:i
7443:d
7438:|
7430:|
7426:F
7422:|
7401:,
7398:)
7389:V
7385:(
7381:m
7378:i
7375:d
7368:)
7365:V
7362:(
7358:m
7355:i
7352:d
7324:,
7321:)
7317:|
7313:F
7309:|
7305:,
7301:|
7297:)
7288:V
7284:(
7280:m
7277:i
7274:d
7269:|
7265:(
7261:x
7258:a
7255:m
7251:=
7247:|
7237:V
7232:|
7228:=
7222:|
7218:A
7214:|
7208:|
7203:F
7199:|
7191:|
7187:A
7183:|
7179:=
7176:)
7173:V
7170:(
7166:m
7163:i
7160:d
7132:F
7112:V
7058:A
7054:F
7047:F
7042:A
7019:F
7013:A
6989:)
6985:F
6980:A
6965:(
6951:V
6927:A
6907:F
6881:A
6877:F
6853:.
6850:F
6845:A
6826:0
6822:)
6816:A
6812:F
6808:(
6802:V
6779:A
6759:F
6733:0
6729:)
6723:A
6719:F
6715:(
6666:f
6642:.
6623:f
6617:A
6603:=
6600:)
6591:e
6587:(
6584:T
6575:f
6569:A
6555:=
6551:)
6540:e
6529:f
6523:A
6508:(
6504:T
6481:V
6461:T
6431:=
6428:)
6422:(
6399:F
6393:A
6390::
6367:V
6347:)
6337:e
6332:(
6329:T
6326:=
6296:V
6276:T
6256:F
6236:A
6210:A
6206:F
6185:V
6162:V
6156:v
6136:f
6116:V
6086:e
6075:f
6069:A
6035:f
6015:A
5989:)
5983:(
5980:f
5977:=
5968:f
5947:F
5941:A
5938::
5935:f
5913:0
5909:)
5903:A
5899:F
5895:(
5875:V
5855:}
5852:A
5843::
5833:e
5828:{
5808:F
5788:V
5762:N
5756:R
5723:R
5695:n
5691:x
5667:,
5662:n
5658:x
5652:n
5648:a
5642:n
5607:R
5585:)
5580:n
5576:x
5572:(
5552:)
5547:n
5543:a
5539:(
5512:N
5506:R
5477:R
5455:i
5433:i
5428:e
5405:N
5398:i
5377:N
5345:R
5320:A
5306:(
5287:e
5261:A
5234:e
5209:V
5181:V
5155:V
5130:V
5103:V
5082:V
5056:V
5027:V
5004:M
4984:n
4978:1
4958:M
4938:y
4935:=
4932:x
4929:M
4909:1
4903:1
4883:y
4863:1
4857:n
4837:x
4817:M
4797:y
4777:x
4757:n
4731:n
4726:R
4704:n
4677:n
4655:n
4650:R
4613:,
4584:,
4579:i
4574:e
4564:i
4559:e
4554:,
4550:x
4541:i
4533:=
4528:i
4523:e
4513:i
4508:e
4503:,
4499:x
4490:i
4482:=
4478:x
4454:V
4447:x
4426:n
4400:n
4396:I
4372:,
4367:n
4363:I
4359:=
4356:E
4347:T
4336:E
4309:]
4304:n
4299:e
4293:|
4285:|
4279:1
4274:e
4269:[
4266:=
4257:E
4234:]
4229:n
4224:e
4218:|
4210:|
4204:1
4199:e
4194:[
4191:=
4188:E
4166:n
4161:R
4139:V
4116:y
4113:+
4110:x
4104:=
4101:)
4098:y
4095:,
4092:x
4089:(
4084:2
4079:e
4057:x
4054:2
4051:=
4048:)
4045:y
4042:,
4039:x
4036:(
4031:1
4026:e
4002:2
3997:e
3973:1
3968:e
3946:}
3943:)
3940:1
3937:,
3934:1
3928:(
3925:=
3920:2
3915:e
3910:,
3907:)
3904:0
3901:,
3898:2
3895:(
3892:=
3887:1
3882:e
3877:{
3854:.
3849:]
3843:1
3838:0
3831:0
3826:1
3820:[
3815:=
3810:]
3798:e
3786:e
3772:e
3760:e
3753:[
3746:]
3734:e
3722:e
3708:e
3696:e
3689:[
3664:1
3661:=
3658:)
3653:2
3648:e
3643:(
3638:2
3633:e
3611:0
3608:=
3605:)
3600:1
3595:e
3590:(
3585:2
3580:e
3558:0
3555:=
3552:)
3547:2
3542:e
3537:(
3532:1
3527:e
3505:1
3502:=
3499:)
3494:1
3489:e
3484:(
3479:1
3474:e
3446:2
3441:e
3417:1
3412:e
3390:}
3387:)
3384:1
3381:,
3378:0
3375:(
3372:=
3367:2
3362:e
3357:,
3354:)
3351:2
3347:/
3343:1
3340:,
3337:2
3333:/
3329:1
3326:(
3323:=
3318:1
3313:e
3308:{
3286:2
3281:R
3259:V
3225:V
3198:V
3177:}
3172:n
3167:e
3162:,
3156:,
3151:1
3146:e
3141:{
3118:)
3113:n
3108:e
3103:(
3100:g
3097:)
3094:x
3091:(
3086:n
3081:e
3076:+
3070:+
3067:)
3062:1
3057:e
3052:(
3049:g
3046:)
3043:x
3040:(
3035:1
3030:e
3025:=
3022:)
3017:n
3012:e
3007:(
3004:g
2999:n
2991:+
2985:+
2982:)
2977:1
2972:e
2967:(
2964:g
2959:1
2951:=
2948:)
2943:n
2938:e
2931:n
2923:+
2917:+
2912:1
2907:e
2900:1
2892:(
2889:g
2886:=
2883:)
2880:x
2877:(
2874:g
2844:V
2837:g
2826:.
2808:V
2787:}
2782:n
2777:e
2772:,
2766:,
2761:1
2756:e
2751:{
2729:i
2702:i
2697:e
2675:0
2672:=
2667:n
2659:=
2653:=
2648:2
2640:=
2635:1
2610:V
2584:V
2577:0
2574:=
2569:n
2564:e
2557:n
2549:+
2543:+
2538:1
2533:e
2526:1
2510:.
2498:n
2495:,
2489:,
2486:2
2483:,
2480:1
2477:=
2474:i
2448:V
2439:i
2434:e
2412:)
2409:y
2406:(
2401:i
2396:e
2388:+
2385:)
2382:x
2379:(
2374:i
2369:e
2364:=
2359:i
2348:+
2343:i
2335:=
2332:)
2329:y
2323:+
2320:x
2317:(
2312:i
2307:e
2283:n
2278:e
2273:)
2268:n
2257:+
2252:n
2244:(
2241:+
2235:+
2230:1
2225:e
2220:)
2215:1
2204:+
2199:1
2191:(
2188:=
2185:y
2179:+
2176:x
2154:i
2146:=
2143:)
2140:y
2137:(
2132:i
2127:e
2103:i
2095:=
2092:)
2089:x
2086:(
2081:i
2076:e
2052:n
2047:e
2040:n
2032:+
2026:+
2021:1
2016:e
2009:1
2001:=
1998:y
1976:n
1971:e
1964:n
1956:+
1950:+
1945:1
1940:e
1933:1
1925:=
1922:x
1902:V
1896:y
1893:,
1890:x
1870:,
1867:n
1864:,
1858:,
1855:2
1852:,
1849:1
1846:=
1843:i
1840:,
1835:i
1830:e
1798:V
1774:n
1771:,
1765:,
1762:1
1759:=
1756:i
1752:,
1747:i
1743:c
1739:=
1736:)
1731:n
1726:e
1719:n
1715:c
1711:+
1705:+
1700:1
1695:e
1688:1
1684:c
1680:(
1675:i
1670:e
1646:}
1641:n
1636:e
1631:,
1625:,
1620:1
1615:e
1610:{
1590:}
1585:n
1580:e
1575:,
1569:,
1564:1
1559:e
1554:{
1509:i
1504:j
1474:i
1469:j
1461:=
1458:)
1453:j
1448:e
1443:(
1438:i
1433:e
1408:F
1400:i
1396:c
1372:n
1369:,
1363:,
1360:1
1357:=
1354:i
1350:,
1345:i
1341:c
1337:=
1334:)
1329:n
1324:e
1317:n
1313:c
1309:+
1303:+
1298:1
1293:e
1286:1
1282:c
1278:(
1273:i
1268:e
1243:V
1223:}
1218:n
1213:e
1208:,
1202:,
1197:1
1192:e
1187:{
1157:V
1136:V
1116:}
1111:n
1106:e
1101:,
1095:,
1090:1
1085:e
1080:{
1057:V
1031:V
1010:V
972:F
960:V
953:V
950::
941:,
905:,
902:x
896:=
893:)
890:x
887:(
862:]
856:,
853:x
850:[
847:=
844:)
841:x
838:(
813:V
791:x
764:V
698:V
674:F
668:a
647:V
641:x
614:V
604:,
573:)
569:)
566:x
563:(
556:(
552:a
549:=
542:)
539:x
536:(
533:)
527:a
524:(
517:)
514:x
511:(
505:+
502:)
499:x
496:(
490:=
483:)
480:x
477:(
474:)
468:+
462:(
434:F
407:V
386:)
383:F
380:,
377:V
374:(
352:(
339:F
333:V
330::
299:V
272:V
245:V
221:F
198:V
85:,
82:V
48:V
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.