4843:
3007:
In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect
2304:
2664:
and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
620:
2458:
1990:
2564:
2146:
is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the
734:
1183:
is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing
3204:
2214:
495:
2389:
1876:
1582:). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a
288:
2501:
1126:
1050:
1830:). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when
767:
1152:
1076:
655:
1586:(for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example
3101:
1449:, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the
1008:. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
4732:
1578:
of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is
4214:
2996:. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs,
4568:
4395:
4558:
1815:
then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
4685:
4540:
4516:
624:
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all
4286:
4259:
4197:
4120:
4091:
4073:
2299:{\displaystyle W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.}
4408:
4108:
615:{\displaystyle B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.}
4497:
4388:
4326:
4304:
4226:
4150:
4031:
4767:
4278:
2478:
4412:
4563:
4352:
4318:
4019:
4895:
4846:
4619:
4553:
4381:
3883:
4583:
4347:
4165:
2453:{\displaystyle B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
4828:
4782:
4706:
4588:
4251:
4170:
4272:
1985:{\displaystyle B^{+}={\tfrac {1}{2}}(B+{}^{\text{t}}B)\qquad B^{-}={\tfrac {1}{2}}(B-{}^{\text{t}}B),}
264:
4880:
4823:
4639:
1249:
3070:
4675:
4573:
4476:
4342:
1800:
1446:
799:
352:
4890:
4885:
4772:
4548:
3062:
1617:
2155:
of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector
4803:
4747:
4711:
3848:
1807:
is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if
1567:
1374:
respectively; they are the vectors orthogonal to the whole space on the left and on the right.
356:
4177:
1090:
1014:
4510:
4160:
2208:
1873:
then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
1827:
1382:
739:
4506:
1131:
1055:
4786:
3326:
2574:
294:
66:
53:
4373:
4236:
4130:
4041:
8:
4752:
4690:
4404:
3853:
2352:
1579:
1165:
The corresponding notion for a module over a commutative ring is that a bilinear form is
306:
58:
2684:, this correspondence between quadratic forms and symmetric bilinear forms breaks down.
4864:
4777:
4644:
3319:
2693:
1583:
1563:
625:
298:
3008:
pairing. A pairing can be nondegenerate without being a perfect pairing, for instance
4757:
4322:
4300:
4282:
4255:
4222:
4193:
4146:
4116:
4087:
4069:
4027:
3909:
3878:
3066:
966:
770:
326:
4762:
4680:
4649:
4629:
4614:
4609:
4604:
4268:
4232:
4126:
4048:
4037:
3863:
3054:
discusses "eight types of inner product". To define them he uses diagonal matrices
2950:
1865:
are equal, and skew-symmetric if and only if they are negatives of one another. If
1823:
1668:
1353:
330:
4441:
2559:{\displaystyle B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.}
4624:
4578:
4526:
4521:
4492:
4243:
4218:
4112:
4023:
3569:
3282:
3051:
2886:
1005:
965:
where the dot ( ⋅ ) indicates the slot into which the argument for the resulting
4451:
3243:
Some of the real symmetric cases are very important. The positive definite case
4813:
4665:
4466:
4142:
4137:
Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner
Product Spaces",
4100:
3873:
2907:
2697:
2598:
1819:
317:
4874:
4818:
4742:
4471:
4456:
4446:
3888:
3061:
having only +1 or −1 for non-zero elements. Some of the "inner products" are
2199:
is nonsingular, and thus if and only if the bilinear form is nondegenerate.
4808:
4461:
4431:
4182:
3843:
45:
28:
4737:
4727:
4634:
4436:
4206:
3868:
3858:
3824:
3338:
3090:
1575:
258:
20:
729:{\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},}
4860:
4670:
4502:
4365:
2835:
819:
302:
89:
4360:
2773:
is a linear map the corresponding bilinear form is given by composing
4065:
3199:{\displaystyle \sum _{k=1}^{p}x_{k}y_{k}-\sum _{k=p+1}^{n}x_{k}y_{k}}
2319:
2877:
Likewise, symmetric bilinear forms may be thought of as elements of
1607:
970:
3239:. Then he articulates the connection to traditional terminology:
2167:, is in the radical of a bilinear form with matrix representation
2007:
2308:
For a non-degenerate form on a finite-dimensional space, the map
2700:, there is a canonical correspondence between bilinear forms on
3050:
Terminology varies in coverage of bilinear forms. For example,
2953:
from two vector spaces over the same base field to that field
3955:
3037:
is nondegenerate, but induces multiplication by 2 on the map
1202:
is nondegenerate but not unimodular, as the induced map from
293:
The definition of a bilinear form can be extended to include
4060:
Cooperstein, Bruce (2010), "Ch 8: Bilinear Forms and Maps",
776:. Then, the matrix of the bilinear form on the new basis is
4403:
4250:, Cambridge Studies in Advanced Mathematics, vol. 50,
1818:
A bilinear form is symmetric (respectively skew-symmetric)
4315:
Principal
Structures and Methods of Representation Theory
2660:
is determined by the symmetric part of the bilinear form
4267:
998:
is an isomorphism, then both are, and the bilinear form
4295:
Shilov, Georgi E. (1977), Silverman, Richard A. (ed.),
3931:
4859:
This article incorporates material from
Unimodular on
3991:
3919:
3910:"Chapter 3. Bilinear forms — Lecture notes for MA1212"
2343:
1944:
1894:
3104:
2848:, so bilinear forms may be thought of as elements of
2504:
2392:
2217:
1879:
1841:
A bilinear form is symmetric if and only if the maps
1134:
1093:
1058:
1017:
742:
658:
498:
267:
2864:
is finite-dimensional) is canonically isomorphic to
4192:, vol. I (2nd ed.), Courier Corporation,
3979:
3967:
1574:, a bilinear form is degenerate if and only if the
4733:Spectral theory of ordinary differential equations
3198:
2944:
2558:
2452:
2298:
1984:
1146:
1120:
1070:
1044:
761:
728:
614:
282:
4014:Adkins, William A.; Weintraub, Steven H. (1992),
1822:its coordinate matrix (relative to any basis) is
70:). In other words, a bilinear form is a function
4872:
4865:Creative Commons Attribution/Share-Alike License
4215:Ergebnisse der Mathematik und ihrer Grenzgebiete
4205:
3943:
2976:Here we still have induced linear mappings from
2895:) and alternating bilinear forms as elements of
1608:Symmetric, skew-symmetric, and alternating forms
793:
4013:
3961:
2008:Reflexive bilinear forms and orthogonal vectors
1549:This form will be nondegenerate if and only if
473:with respect to this basis, and similarly, the
2687:
329:, which are similar to bilinear forms but are
4389:
2568:
1229:is finite-dimensional then one can identify
336:
4317:, Translations of Mathematical Monographs,
4059:
2193:. It is trivial if and only if the matrix
4396:
4382:
4312:
4248:Clifford Algebras and the Classical Groups
3937:
2728:the corresponding linear map is given by
2465:A bilinear form on a normed vector space
1764:Every alternating form is skew-symmetric.
532:
270:
4686:Group algebra of a locally compact group
4242:
4187:
4047:
3997:
3925:
3280:, then Lorentzian space is also called
4873:
4313:Zhelobenko, Dmitriĭ Petrovich (2006),
4294:
4136:
4099:
4016:Algebra: An Approach via Module Theory
3985:
3973:
2949:Much of the theory is available for a
2187:. The radical is always a subspace of
976:For a finite-dimensional vector space
4377:
4158:
4081:
3949:
3258:, while the case of a single minus,
3098:are spelled out. The bilinear form
2344:Bounded and elliptic bilinear forms
814:defines a pair of linear maps from
13:
4109:Undergraduate Texts in Mathematics
3313:
3077:, the instances with real numbers
2939:
325:, one is often more interested in
290:is an example of a bilinear form.
64:(the elements of which are called
51:(the elements of which are called
14:
4907:
4335:
2575:Quadratic form § Definitions
1309:to be the bilinear form given by
4842:
4841:
4768:Topological quantum field theory
4105:Finite-dimensional vector spaces
2539:
2520:
2512:
2439:
2426:
2408:
2400:
2278:
2259:
2251:
2237:
1612:We define a bilinear form to be
713:
661:
542:
526:
514:
506:
283:{\displaystyle \mathbb {R} ^{n}}
2945:Pairs of distinct vector spaces
1929:
1512:one can obtain a bilinear form
1381:is finite-dimensional then the
16:Scalar-valued bilinear function
4863:, which is licensed under the
3902:
3073:. Rather than a general field
2543:
2535:
2524:
2508:
2443:
2435:
2430:
2422:
2412:
2396:
2263:
2247:
2103:be a reflexive bilinear form.
1976:
1955:
1926:
1905:
1770:This can be seen by expanding
1109:
1097:
1033:
1021:
518:
502:
1:
4564:Uniform boundedness principle
4319:American Mathematical Society
4020:Graduate Texts in Mathematics
4007:
2597:, there exists an associated
2161:, with matrix representation
1429:are linear isomorphisms from
1403:. If this number is equal to
1265:is infinite-dimensional then
794:Non-degenerate bilinear forms
788:
92:in each argument separately:
3895:
3884:System of bilinear equations
3725:Conversely, a bilinear form
7:
4348:Encyclopedia of Mathematics
4274:Linear Algebra and Geometry
4166:Encyclopedia of Mathematics
4163:, in Hazewinkel, M. (ed.),
3962:Adkins & Weintraub 1992
3836:
3305:will be referred to as the
2820:The set of all linear maps
2755:In the other direction, if
2688:Relation to tensor products
2206:is a subspace. Define the
2117:orthogonal with respect to
1283:restricted to the image of
429:matrix of the bilinear form
10:
4912:
4707:Invariant subspace problem
4252:Cambridge University Press
4171:Kluwer Academic Publishers
3683:induces the bilinear form
3617:induces the bilinear form
2572:
797:
486:represents another vector
4837:
4796:
4720:
4699:
4658:
4597:
4539:
4485:
4427:
4420:
4209:; Husemoller, D. (1973),
4188:Jacobson, Nathan (2009),
2569:Associated quadratic form
2481:, if there is a constant
2366:, if there is a constant
1445:is nondegenerate. By the
1239:. One can then show that
916:This is often denoted as
337:Coordinate representation
4676:Spectrum of a C*-algebra
4271:; A. O. Remizov (2012),
4211:Symmetric Bilinear Forms
4139:Spinors and calibrations
4082:Grove, Larry C. (1997),
1394:is equal to the rank of
1222:is multiplication by 2.
1121:{\displaystyle B(x,y)=0}
1045:{\displaystyle B(x,y)=0}
800:Degenerate bilinear form
4773:Noncommutative geometry
4062:Advanced Linear Algebra
3575:canonical bilinear form
2322:, and the dimension of
1566:then, relative to some
762:{\displaystyle S_{i,j}}
4829:Tomita–Takesaki theory
4804:Approximation property
4748:Calculus of variations
4086:, Wiley-Interscience,
3849:Category:Bilinear maps
3311:
3200:
3175:
3125:
2777:with the bilinear map
2724:is a bilinear form on
2579:For any bilinear form
2560:
2454:
2300:
1986:
1148:
1147:{\displaystyle x\in V}
1122:
1072:
1071:{\displaystyle y\in V}
1046:
763:
730:
694:
616:
575:
284:
4824:Banach–Mazur distance
4787:Generalized functions
4159:Popov, V. L. (1987),
4084:Groups and characters
3241:
3201:
3149:
3105:
2656:, the quadratic form
2573:Further information:
2561:
2455:
2351:A bilinear form on a
2301:
2209:orthogonal complement
1987:
1498:Given any linear map
1233:with its double dual
1149:
1123:
1073:
1047:
969:is to be placed (see
798:Further information:
764:
731:
674:
628:. More precisely, if
617:
549:
285:
4569:Kakutani fixed-point
4554:Riesz representation
4111:, Berlin, New York:
3102:
2885:(dual of the second
2502:
2390:
2215:
1998:is the transpose of
1877:
1447:rank–nullity theorem
1274:is the transpose of
1132:
1091:
1056:
1015:
804:Every bilinear form
740:
656:
648:is another basis of
496:
467:represents a vector
307:module homomorphisms
265:
4896:Multilinear algebra
4753:Functional calculus
4712:Mahler's conjecture
4691:Von Neumann algebra
4405:Functional analysis
4068:, pp. 249–88,
3854:Inner product space
3656:, and a linear map
3290:. The special case
3288:Minkowski spacetime
3208:real symmetric case
2353:normed vector space
2274: for all
1604:over the integers.
1555:is an isomorphism.
1299:one can define the
4778:Riemann hypothesis
4477:Topological vector
4269:Shafarevich, I. R.
4173:, pp. 390–392
4145:, pp. 19–40,
3573:, also called the
3196:
3083:, complex numbers
3067:sesquilinear forms
2694:universal property
2556:
2488:such that for all
2450:
2372:such that for all
2296:
1982:
1953:
1903:
1564:finite-dimensional
1453:of nondegeneracy:
1252:of the linear map
1144:
1118:
1068:
1042:
759:
726:
612:
355:vector space with
327:sesquilinear forms
280:
223:
219:
143:
139:
4855:
4854:
4758:Integral operator
4535:
4534:
4288:978-3-642-30993-9
4261:978-0-521-55177-9
4199:978-0-486-47189-1
4122:978-0-387-90093-3
4093:978-0-471-16340-4
4075:978-1-4398-2966-0
4022:, vol. 136,
3879:Sesquilinear form
2275:
2173:, if and only if
2004:(defined above).
1970:
1952:
1920:
1902:
967:linear functional
771:invertible matrix
534:
333:in one argument.
4903:
4881:Abstract algebra
4845:
4844:
4763:Jones polynomial
4681:Operator algebra
4425:
4424:
4398:
4391:
4384:
4375:
4374:
4370:
4356:
4331:
4309:
4291:
4264:
4244:Porteous, Ian R.
4239:
4217:, vol. 73,
4202:
4174:
4155:
4133:
4096:
4078:
4056:
4044:
4001:
3995:
3989:
3983:
3977:
3971:
3965:
3959:
3953:
3947:
3941:
3935:
3929:
3923:
3917:
3916:
3914:
3906:
3864:Multilinear form
3832:
3822:
3816:
3781:
3742:
3721:
3682:
3655:
3616:
3586:
3567:is known as the
3566:
3531:
3517:
3503:
3487:
3456:
3412:
3363:
3345:
3336:
3329:
3324:
3304:
3279:
3271:Lorentzian space
3268:
3253:
3238:
3224:
3205:
3203:
3202:
3197:
3195:
3194:
3185:
3184:
3174:
3169:
3145:
3144:
3135:
3134:
3124:
3119:
3097:
3088:
3082:
3076:
3063:symplectic forms
3046:
3036:
3021:
3000:is said to be a
2995:
2989:
2985:
2979:
2972:
2951:bilinear mapping
2935:
2923:
2915:
2905:
2894:
2884:
2873:
2863:
2859:
2847:
2833:
2816:
2806:
2794:
2772:
2752:
2727:
2723:
2717:
2704:and linear maps
2703:
2683:
2675:
2655:
2644:
2613:
2596:
2565:
2563:
2562:
2557:
2552:
2551:
2546:
2542:
2523:
2515:
2497:
2487:
2472:
2459:
2457:
2456:
2451:
2446:
2442:
2433:
2429:
2411:
2403:
2385:
2371:
2361:
2339:
2327:
2317:
2305:
2303:
2302:
2297:
2292:
2288:
2281:
2276:
2273:
2262:
2254:
2240:
2227:
2226:
2205:
2198:
2192:
2186:
2172:
2166:
2160:
2145:
2140:A bilinear form
2136:
2102:
2066:
2051:
2032:
2015:A bilinear form
2003:
1997:
1991:
1989:
1988:
1983:
1972:
1971:
1968:
1966:
1954:
1945:
1939:
1938:
1922:
1921:
1918:
1916:
1904:
1895:
1889:
1888:
1872:
1864:
1837:
1814:
1806:
1792:
1757:
1753:
1747:
1741:
1713:
1712:
1705:
1704:
1696:
1692:
1686:
1663:
1659:
1653:
1647:
1603:
1573:
1561:
1554:
1511:
1494:
1480:
1444:
1439:. In this case
1438:
1432:
1428:
1419:
1410:
1402:
1393:
1380:
1373:
1364:
1351:
1308:
1298:
1292:
1286:
1282:
1273:
1264:
1260:
1247:
1238:
1232:
1228:
1221:
1211:
1201:
1182:
1171:
1170:
1160:
1153:
1151:
1150:
1145:
1127:
1125:
1124:
1119:
1084:
1077:
1075:
1074:
1069:
1051:
1049:
1048:
1043:
1003:
997:
988:
979:
962:
939:
913:
882:
850:
826:
817:
813:
809:
784:
775:
768:
766:
765:
760:
758:
757:
735:
733:
732:
727:
722:
721:
716:
710:
709:
693:
688:
670:
669:
664:
651:
647:
621:
619:
618:
613:
608:
607:
598:
597:
585:
584:
574:
569:
545:
537:
536:
535:
529:
517:
509:
491:
485:
479:
472:
466:
460:
450:
426:
390:
377:
350:
346:
331:conjugate linear
324:
316:is the field of
315:
289:
287:
286:
281:
279:
278:
273:
253:
224:
220:
217:
173:
144:
140:
137:
87:
50:
43:
4911:
4910:
4906:
4905:
4904:
4902:
4901:
4900:
4871:
4870:
4856:
4851:
4833:
4797:Advanced topics
4792:
4716:
4695:
4654:
4620:Hilbert–Schmidt
4593:
4584:Gelfand–Naimark
4531:
4481:
4416:
4402:
4361:"Bilinear form"
4359:
4343:"Bilinear form"
4341:
4338:
4329:
4307:
4289:
4262:
4229:
4219:Springer-Verlag
4200:
4169:, vol. 1,
4161:"Bilinear form"
4153:
4123:
4113:Springer-Verlag
4101:Halmos, Paul R.
4094:
4076:
4034:
4024:Springer-Verlag
4010:
4005:
4004:
3996:
3992:
3984:
3980:
3972:
3968:
3960:
3956:
3948:
3944:
3938:Zhelobenko 2006
3936:
3932:
3924:
3920:
3912:
3908:
3907:
3903:
3898:
3893:
3839:
3828:
3818:
3783:
3748:
3726:
3684:
3657:
3618:
3591:
3578:
3570:natural pairing
3536:
3519:
3505:
3491:
3488:
3459:
3457:
3415:
3413:
3371:
3347:
3341:
3332:
3327:
3322:
3316:
3314:General modules
3291:
3283:Minkowski space
3274:
3259:
3256:Euclidean space
3244:
3226:
3211:
3190:
3186:
3180:
3176:
3170:
3153:
3140:
3136:
3130:
3126:
3120:
3109:
3103:
3100:
3099:
3093:
3084:
3078:
3074:
3071:Hermitian forms
3059:
3052:F. Reese Harvey
3038:
3023:
3009:
3002:perfect pairing
2991:
2987:
2981:
2977:
2974:
2956:
2947:
2942:
2940:Generalizations
2925:
2917:
2911:
2896:
2890:
2887:symmetric power
2878:
2865:
2861:
2849:
2839:
2821:
2808:
2796:
2778:
2756:
2753:
2731:
2725:
2719:
2705:
2701:
2690:
2677:
2669:
2649:
2615:
2601:
2580:
2577:
2571:
2547:
2538:
2534:
2533:
2519:
2511:
2503:
2500:
2499:
2489:
2482:
2466:
2438:
2434:
2425:
2421:
2407:
2399:
2391:
2388:
2387:
2373:
2367:
2355:
2346:
2329:
2323:
2309:
2277:
2272:
2258:
2250:
2236:
2235:
2231:
2222:
2218:
2216:
2213:
2212:
2203:
2194:
2188:
2174:
2168:
2162:
2156:
2141:
2138:
2123:
2086:
2080:
2053:
2038:
2016:
2010:
1999:
1993:
1967:
1965:
1964:
1943:
1934:
1930:
1917:
1915:
1914:
1893:
1884:
1880:
1878:
1875:
1874:
1866:
1855:
1848:
1842:
1831:
1808:
1804:
1771:
1755:
1749:
1743:
1716:
1710:
1709:
1702:
1701:
1694:
1688:
1673:
1661:
1655:
1649:
1622:
1610:
1587:
1571:
1559:
1550:
1547:
1499:
1496:
1486:
1467:
1440:
1434:
1430:
1427:
1421:
1418:
1412:
1404:
1401:
1395:
1392:
1386:
1378:
1372:
1366:
1363:
1357:
1347:
1336:
1304:
1294:
1288:
1284:
1281:
1275:
1272:
1266:
1262:
1259:
1253:
1246:
1240:
1234:
1230:
1226:
1213:
1203:
1185:
1174:
1168:
1167:
1155:
1133:
1130:
1129:
1092:
1089:
1088:
1079:
1057:
1054:
1053:
1016:
1013:
1012:
999:
996:
990:
987:
981:
980:, if either of
977:
963:
948:
942:
940:
925:
919:
914:
891:
885:
883:
860:
854:
841:
834:
828:
822:
815:
811:
805:
802:
796:
791:
777:
773:
747:
743:
741:
738:
737:
717:
712:
711:
699:
695:
689:
678:
665:
660:
659:
657:
654:
653:
649:
645:
636:
629:
603:
599:
590:
586:
580:
576:
570:
553:
541:
531:
530:
525:
524:
513:
505:
497:
494:
493:
487:
481:
474:
468:
462:
455:
448:
439:
432:
424:
415:
401:
396:
382:
375:
366:
359:
348:
342:
339:
320:
318:complex numbers
313:
274:
269:
268:
266:
263:
262:
225:
222:
218:
176:
145:
142:
138:
96:
71:
48:
31:
17:
12:
11:
5:
4909:
4899:
4898:
4893:
4891:Linear algebra
4888:
4886:Bilinear forms
4883:
4853:
4852:
4850:
4849:
4838:
4835:
4834:
4832:
4831:
4826:
4821:
4816:
4814:Choquet theory
4811:
4806:
4800:
4798:
4794:
4793:
4791:
4790:
4780:
4775:
4770:
4765:
4760:
4755:
4750:
4745:
4740:
4735:
4730:
4724:
4722:
4718:
4717:
4715:
4714:
4709:
4703:
4701:
4697:
4696:
4694:
4693:
4688:
4683:
4678:
4673:
4668:
4666:Banach algebra
4662:
4660:
4656:
4655:
4653:
4652:
4647:
4642:
4637:
4632:
4627:
4622:
4617:
4612:
4607:
4601:
4599:
4595:
4594:
4592:
4591:
4589:Banach–Alaoglu
4586:
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4545:
4543:
4537:
4536:
4533:
4532:
4530:
4529:
4524:
4519:
4517:Locally convex
4514:
4500:
4495:
4489:
4487:
4483:
4482:
4480:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4444:
4439:
4434:
4428:
4422:
4418:
4417:
4401:
4400:
4393:
4386:
4378:
4372:
4371:
4357:
4337:
4336:External links
4334:
4333:
4332:
4327:
4310:
4305:
4297:Linear Algebra
4292:
4287:
4265:
4260:
4240:
4227:
4203:
4198:
4185:
4156:
4151:
4143:Academic Press
4134:
4121:
4097:
4092:
4079:
4074:
4057:
4045:
4032:
4009:
4006:
4003:
4002:
4000:, p. 233.
3990:
3978:
3966:
3964:, p. 359.
3954:
3942:
3930:
3928:, p. 346.
3918:
3900:
3899:
3897:
3894:
3892:
3891:
3886:
3881:
3876:
3874:Quadratic form
3871:
3866:
3861:
3856:
3851:
3846:
3840:
3838:
3835:
3458:
3414:
3370:
3315:
3312:
3206:is called the
3193:
3189:
3183:
3179:
3173:
3168:
3165:
3162:
3159:
3156:
3152:
3148:
3143:
3139:
3133:
3129:
3123:
3118:
3115:
3112:
3108:
3057:
2955:
2946:
2943:
2941:
2938:
2908:exterior power
2730:
2698:tensor product
2689:
2686:
2599:quadratic form
2570:
2567:
2555:
2550:
2545:
2541:
2537:
2532:
2529:
2526:
2522:
2518:
2514:
2510:
2507:
2449:
2445:
2441:
2437:
2432:
2428:
2424:
2420:
2417:
2414:
2410:
2406:
2402:
2398:
2395:
2345:
2342:
2295:
2291:
2287:
2284:
2280:
2271:
2268:
2265:
2261:
2257:
2253:
2249:
2246:
2243:
2239:
2234:
2230:
2225:
2221:
2081:
2011:
2009:
2006:
1981:
1978:
1975:
1963:
1960:
1957:
1951:
1948:
1942:
1937:
1933:
1928:
1925:
1913:
1910:
1907:
1901:
1898:
1892:
1887:
1883:
1853:
1846:
1828:skew-symmetric
1826:(respectively
1820:if and only if
1801:characteristic
1797:
1796:
1795:
1794:
1768:
1765:
1762:
1703:skew-symmetric
1698:
1665:
1609:
1606:
1522:
1455:
1425:
1416:
1399:
1390:
1370:
1361:
1311:
1279:
1270:
1257:
1244:
1163:
1162:
1143:
1140:
1137:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1086:
1067:
1064:
1061:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1004:is said to be
994:
985:
946:
941:
923:
918:
889:
884:
858:
853:
839:
832:
795:
792:
790:
787:
756:
753:
750:
746:
725:
720:
715:
708:
705:
702:
698:
692:
687:
684:
681:
677:
673:
668:
663:
641:
634:
611:
606:
602:
596:
593:
589:
583:
579:
573:
568:
565:
562:
559:
556:
552:
548:
544:
540:
528:
523:
520:
516:
512:
508:
504:
501:
444:
437:
427:is called the
420:
411:
399:
371:
364:
338:
335:
277:
272:
255:
254:
174:
15:
9:
6:
4:
3:
2:
4908:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4878:
4876:
4869:
4868:
4866:
4862:
4848:
4840:
4839:
4836:
4830:
4827:
4825:
4822:
4820:
4819:Weak topology
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4801:
4799:
4795:
4788:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4749:
4746:
4744:
4743:Index theorem
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4725:
4723:
4719:
4713:
4710:
4708:
4705:
4704:
4702:
4700:Open problems
4698:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4663:
4661:
4657:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4602:
4600:
4596:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4546:
4544:
4542:
4538:
4528:
4525:
4523:
4520:
4518:
4515:
4512:
4508:
4504:
4501:
4499:
4496:
4494:
4491:
4490:
4488:
4484:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4429:
4426:
4423:
4419:
4414:
4410:
4406:
4399:
4394:
4392:
4387:
4385:
4380:
4379:
4376:
4368:
4367:
4362:
4358:
4354:
4350:
4349:
4344:
4340:
4339:
4330:
4328:0-8218-3731-1
4324:
4320:
4316:
4311:
4308:
4306:0-486-63518-X
4302:
4298:
4293:
4290:
4284:
4280:
4276:
4275:
4270:
4266:
4263:
4257:
4253:
4249:
4245:
4241:
4238:
4234:
4230:
4228:3-540-06009-X
4224:
4220:
4216:
4212:
4208:
4204:
4201:
4195:
4191:
4190:Basic Algebra
4186:
4184:
4181:, p. 390, at
4180:
4179:
4178:Bilinear form
4172:
4168:
4167:
4162:
4157:
4154:
4152:0-12-329650-1
4148:
4144:
4140:
4135:
4132:
4128:
4124:
4118:
4114:
4110:
4106:
4102:
4098:
4095:
4089:
4085:
4080:
4077:
4071:
4067:
4063:
4058:
4054:
4050:
4046:
4043:
4039:
4035:
4033:3-540-97839-9
4029:
4025:
4021:
4017:
4012:
4011:
3999:
3998:Bourbaki 1970
3994:
3988:, p. 23.
3987:
3982:
3976:, p. 22.
3975:
3970:
3963:
3958:
3951:
3946:
3940:, p. 11.
3939:
3934:
3927:
3926:Jacobson 2009
3922:
3915:. 2021-01-16.
3911:
3905:
3901:
3890:
3889:Metric tensor
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3841:
3834:
3831:
3826:
3821:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3786:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:-linear maps
3746:
3741:
3737:
3733:
3729:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3687:
3680:
3676:
3672:
3668:
3664:
3660:
3653:
3649:
3645:
3641:
3637:
3633:
3629:
3625:
3621:
3614:
3610:
3606:
3602:
3598:
3594:
3590:A linear map
3588:
3585:
3581:
3576:
3572:
3571:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3537:⟨⋅,⋅⟩ :
3533:
3530:
3526:
3522:
3516:
3512:
3508:
3502:
3498:
3494:
3486:
3482:
3478:
3474:
3470:
3466:
3462:
3454:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3422:
3418:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3369:
3367:
3366:bilinear form
3362:
3358:
3354:
3350:
3344:
3340:
3335:
3331:
3321:
3310:
3308:
3302:
3298:
3294:
3289:
3285:
3284:
3277:
3272:
3266:
3262:
3257:
3251:
3247:
3240:
3237:
3233:
3229:
3222:
3218:
3214:
3209:
3191:
3187:
3181:
3177:
3171:
3166:
3163:
3160:
3157:
3154:
3150:
3146:
3141:
3137:
3131:
3127:
3121:
3116:
3113:
3110:
3106:
3096:
3092:
3087:
3081:
3072:
3068:
3065:and some are
3064:
3060:
3053:
3048:
3045:
3041:
3035:
3031:
3027:
3020:
3016:
3012:
3005:
3003:
2999:
2994:
2984:
2971:
2967:
2963:
2959:
2954:
2952:
2937:
2933:
2929:
2921:
2914:
2909:
2904:
2900:
2893:
2888:
2882:
2875:
2872:
2868:
2857:
2853:
2846:
2842:
2837:
2832:
2828:
2824:
2818:
2815:
2811:
2804:
2800:
2793:
2789:
2785:
2781:
2776:
2771:
2767:
2763:
2759:
2750:
2746:
2742:
2738:
2734:
2729:
2722:
2716:
2712:
2708:
2699:
2695:
2685:
2681:
2673:
2666:
2663:
2659:
2653:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2612:
2608:
2604:
2600:
2595:
2591:
2587:
2583:
2576:
2566:
2553:
2548:
2530:
2527:
2516:
2505:
2496:
2492:
2485:
2480:
2476:
2470:
2464:
2460:
2447:
2418:
2415:
2404:
2393:
2384:
2380:
2376:
2370:
2365:
2359:
2354:
2350:
2341:
2337:
2333:
2326:
2321:
2316:
2312:
2306:
2293:
2289:
2285:
2282:
2269:
2266:
2255:
2244:
2241:
2232:
2228:
2223:
2219:
2211:
2210:
2200:
2197:
2191:
2184:
2181:
2177:
2171:
2165:
2159:
2154:
2150:
2144:
2134:
2130:
2126:
2121:
2120:
2114:
2110:
2106:
2101:
2097:
2093:
2089:
2084:
2078:
2074:
2070:
2064:
2060:
2056:
2049:
2045:
2041:
2036:
2031:
2027:
2023:
2019:
2014:
2005:
2002:
1996:
1979:
1973:
1961:
1958:
1949:
1946:
1940:
1935:
1931:
1923:
1911:
1908:
1899:
1896:
1890:
1885:
1881:
1870:
1863:
1859:
1852:
1845:
1839:
1835:
1829:
1825:
1821:
1816:
1812:
1802:
1790:
1786:
1782:
1778:
1774:
1769:
1766:
1763:
1760:
1759:
1752:
1746:
1739:
1735:
1731:
1727:
1723:
1719:
1714:
1711:antisymmetric
1706:
1699:
1691:
1684:
1680:
1676:
1671:
1670:
1666:
1658:
1652:
1645:
1641:
1637:
1633:
1629:
1625:
1620:
1619:
1615:
1614:
1613:
1605:
1602:
1598:
1594:
1590:
1585:
1581:
1577:
1569:
1565:
1556:
1553:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1519:
1515:
1510:
1506:
1502:
1493:
1489:
1484:
1478:
1474:
1470:
1465:
1464:nondegenerate
1461:
1458:
1454:
1452:
1448:
1443:
1437:
1424:
1415:
1408:
1398:
1389:
1384:
1375:
1369:
1360:
1355:
1350:
1345:
1344:right radical
1341:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1307:
1302:
1297:
1291:
1278:
1269:
1256:
1251:
1243:
1237:
1223:
1220:
1216:
1210:
1206:
1200:
1196:
1192:
1188:
1181:
1177:
1172:
1158:
1154:implies that
1141:
1138:
1135:
1115:
1112:
1106:
1103:
1100:
1094:
1087:
1082:
1078:implies that
1065:
1062:
1059:
1039:
1036:
1030:
1027:
1024:
1018:
1011:
1010:
1009:
1007:
1006:nondegenerate
1002:
993:
984:
974:
972:
968:
960:
956:
952:
945:
937:
933:
929:
922:
917:
911:
907:
903:
899:
895:
888:
880:
876:
872:
868:
864:
857:
852:
849:
845:
838:
831:
825:
821:
808:
801:
786:
783:
780:
772:
754:
751:
748:
744:
723:
718:
706:
703:
700:
696:
690:
685:
682:
679:
675:
671:
666:
644:
640:
633:
627:
622:
609:
604:
600:
594:
591:
587:
581:
577:
571:
566:
563:
560:
557:
554:
550:
546:
538:
521:
510:
499:
490:
484:
477:
471:
465:
458:
452:
447:
443:
436:
431:on the basis
430:
423:
419:
414:
410:
406:
402:
395:, defined by
394:
389:
385:
379:
374:
370:
363:
358:
354:
345:
334:
332:
328:
323:
319:
310:
308:
304:
300:
296:
291:
275:
260:
251:
247:
243:
239:
236:
232:
228:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
152:
148:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
94:
93:
91:
86:
82:
78:
74:
69:
68:
63:
60:
56:
55:
47:
42:
38:
34:
30:
26:
25:bilinear form
22:
4858:
4857:
4809:Balanced set
4783:Distribution
4721:Applications
4574:Krein–Milman
4559:Closed graph
4364:
4346:
4314:
4296:
4273:
4247:
4210:
4189:
4183:Google Books
4176:
4164:
4138:
4104:
4083:
4061:
4052:
4049:Bourbaki, N.
4015:
3993:
3981:
3969:
3957:
3945:
3933:
3921:
3904:
3844:Bilinear map
3829:
3823:denotes the
3819:
3812:
3808:
3804:
3800:
3796:
3792:
3788:
3784:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3744:
3743:induces the
3739:
3735:
3731:
3727:
3724:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3685:
3678:
3674:
3670:
3666:
3662:
3658:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3612:
3608:
3604:
3600:
3596:
3592:
3589:
3583:
3579:
3574:
3568:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3535:The mapping
3534:
3528:
3524:
3520:
3514:
3510:
3506:
3500:
3496:
3492:
3489:
3484:
3480:
3476:
3472:
3468:
3464:
3460:
3452:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3365:
3364:is called a
3360:
3356:
3352:
3348:
3346:, a mapping
3342:
3333:
3325:and a right
3317:
3306:
3300:
3296:
3292:
3287:
3281:
3275:
3270:
3264:
3260:
3255:
3249:
3245:
3242:
3235:
3231:
3227:
3220:
3216:
3212:
3210:and labeled
3207:
3094:
3085:
3079:
3055:
3049:
3043:
3039:
3033:
3029:
3025:
3018:
3014:
3010:
3006:
3001:
2997:
2992:
2982:
2975:
2969:
2965:
2961:
2957:
2948:
2931:
2927:
2919:
2912:
2906:(the second
2902:
2898:
2891:
2880:
2876:
2870:
2866:
2860:which (when
2855:
2851:
2844:
2840:
2830:
2826:
2822:
2819:
2813:
2809:
2802:
2798:
2791:
2787:
2783:
2779:
2774:
2769:
2765:
2761:
2757:
2754:
2748:
2744:
2740:
2736:
2732:
2720:
2714:
2710:
2706:
2691:
2679:
2671:
2667:
2661:
2657:
2651:
2647:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2610:
2606:
2602:
2593:
2589:
2585:
2581:
2578:
2494:
2490:
2483:
2474:
2468:
2462:
2461:
2382:
2378:
2374:
2368:
2363:
2357:
2348:
2347:
2335:
2331:
2324:
2314:
2310:
2307:
2207:
2201:
2195:
2189:
2182:
2179:
2175:
2169:
2163:
2157:
2152:
2148:
2142:
2139:
2132:
2128:
2124:
2118:
2116:
2112:
2108:
2104:
2099:
2095:
2091:
2087:
2082:
2076:
2072:
2068:
2062:
2058:
2054:
2047:
2043:
2039:
2034:
2029:
2025:
2021:
2017:
2012:
2000:
1994:
1868:
1861:
1857:
1850:
1843:
1840:
1833:
1817:
1810:
1798:
1788:
1784:
1780:
1776:
1772:
1750:
1744:
1737:
1733:
1729:
1725:
1721:
1717:
1708:
1700:
1689:
1682:
1678:
1674:
1667:
1656:
1650:
1643:
1639:
1635:
1631:
1627:
1623:
1616:
1611:
1600:
1596:
1592:
1588:
1580:non-singular
1557:
1551:
1548:
1543:
1539:
1535:
1531:
1527:
1523:
1517:
1513:
1508:
1504:
1500:
1497:
1491:
1487:
1482:
1476:
1472:
1468:
1463:
1459:
1456:
1450:
1441:
1435:
1422:
1413:
1406:
1396:
1387:
1376:
1367:
1358:
1348:
1346:of the form
1343:
1340:left radical
1339:
1337:
1332:
1328:
1324:
1320:
1316:
1312:
1305:
1300:
1295:
1289:
1276:
1267:
1254:
1241:
1235:
1224:
1218:
1214:
1208:
1204:
1198:
1194:
1190:
1186:
1179:
1175:
1166:
1164:
1156:
1080:
1000:
991:
982:
975:
964:
958:
954:
950:
943:
935:
931:
927:
920:
915:
909:
905:
901:
897:
893:
886:
878:
874:
870:
866:
862:
855:
847:
843:
836:
829:
823:
806:
803:
781:
778:
642:
638:
631:
623:
488:
482:
475:
469:
463:
456:
453:
445:
441:
434:
428:
421:
417:
412:
408:
404:
397:
392:
387:
383:
380:
372:
368:
361:
343:
340:
321:
311:
305:replaced by
292:
256:
249:
245:
241:
237:
234:
230:
226:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
169:
165:
161:
157:
153:
150:
146:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
84:
80:
76:
72:
65:
61:
52:
46:vector space
40:
36:
32:
29:bilinear map
24:
18:
4738:Heat kernel
4728:Hardy space
4635:Trace class
4549:Hahn–Banach
4511:Topological
3986:Harvey 1990
3974:Harvey 1990
3869:Polar space
3859:Linear form
3825:double dual
3339:dual module
3091:quaternions
2986:, and from
2795:that sends
2614:defined by
2463:Definition:
2349:Definition:
2083:Definition:
2013:Definition:
1761:Proposition
1669:alternating
1576:determinant
1457:Definition:
353:dimensional
303:linear maps
259:dot product
21:mathematics
4875:Categories
4861:PlanetMath
4671:C*-algebra
4486:Properties
4366:PlanetMath
4237:0292.10016
4207:Milnor, J.
4131:0288.15002
4055:, Springer
4042:0768.00003
4008:References
3950:Grove 1997
3307:split-case
3269:is called
3254:is called
2836:dual space
2033:is called
1451:definition
1169:unimodular
820:dual space
789:Properties
736:where the
4645:Unbounded
4640:Transpose
4598:Operators
4527:Separable
4522:Reflexive
4507:Algebraic
4493:Barrelled
4353:EMS Press
4299:, Dover,
4066:CRC Press
3896:Citations
3817:. Here,
3787:′ :
3700: : (
3634: : (
3549: : (
3151:∑
3147:−
3107:∑
2528:≥
2416:≤
2320:bijective
2283:∈
2242:∣
2224:⊥
2035:reflexive
1962:−
1936:−
1824:symmetric
1618:symmetric
1301:transpose
1293:). Given
1250:transpose
1139:∈
1063:∈
827:. Define
676:∑
626:congruent
551:∑
57:) over a
4847:Category
4659:Algebras
4541:Theorems
4498:Complete
4467:Schwartz
4413:glossary
4279:Springer
4246:(1995),
4175:. Also:
4103:(1974),
4051:(1970),
3837:See also
3795: :
3760: :
3752: :
3730: :
3688: :
3669: :
3661: :
3622: :
3603: :
3595: :
3518:and all
3490:for all
3351: :
3337:and its
3318:Given a
3225:, where
2960: :
2930:) ≃ Sym(
2760: :
2627: :
2619: :
2605: :
2584: :
2544:‖
2536:‖
2479:coercive
2475:elliptic
2444:‖
2436:‖
2431:‖
2423:‖
2334:) − dim(
2202:Suppose
2090: :
2067:for all
2052:implies
2020: :
1742:for all
1687:for all
1648:for all
1503: :
1485:implies
1481:for all
1352:are the
1128:for all
1052:for all
971:Currying
769:form an
492:, then:
88:that is
75: :
4650:Unitary
4630:Nuclear
4615:Compact
4610:Bounded
4605:Adjoint
4579:Min–max
4472:Sobolev
4457:Nuclear
4447:Hilbert
4442:Fréchet
4407: (
4355:, 2001
4053:Algebra
3330:-module
2834:is the
2696:of the
2692:By the
2364:bounded
2153:radical
2151:or the
1799:If the
1354:kernels
1248:is the
818:to its
652:, then
480:matrix
461:matrix
454:If the
391:matrix
301:, with
297:over a
295:modules
67:scalars
54:vectors
4625:Normal
4462:Orlicz
4452:Hölder
4432:Banach
4421:Spaces
4409:topics
4325:
4303:
4285:
4258:
4235:
4225:
4196:
4149:
4129:
4119:
4090:
4072:
4040:
4030:
3504:, all
3267:−1, 1)
3089:, and
2916:). If
2682:> 1
2486:> 0
2471:, ‖⋅‖)
2360:, ‖⋅‖)
2178:= 0 ⇔
2149:kernel
1992:where
347:be an
90:linear
4437:Besov
3913:(PDF)
3708:) ↦ ⟨
3642:) ↦ ⟨
3273:. If
3032:) ↦ 2
2901:) ≃ Λ
2718:. If
2674:) = 2
2670:char(
2668:When
2654:) ≠ 2
2650:char(
2648:When
2477:, or
2135:) = 0
2065:) = 0
2050:) = 0
1871:) ≠ 2
1867:char(
1836:) ≠ 2
1832:char(
1813:) = 2
1809:char(
1767:Proof
1728:) = −
1685:) = 0
1599:) = 2
1568:basis
1479:) = 0
1411:then
1197:) = 2
637:, …,
440:, …,
367:, …,
357:basis
312:When
59:field
44:on a
27:is a
4785:(or
4503:Dual
4323:ISBN
4301:ISBN
4283:ISBN
4256:ISBN
4223:ISBN
4194:ISBN
4147:ISBN
4117:ISBN
4088:ISBN
4070:ISBN
4028:ISBN
3782:and
3557:) ↦
3471:) =
3443:) +
3431:) =
3399:) +
3387:) =
3320:ring
3252:, 0)
3022:via
2926:(Sym
2918:char
2879:(Sym
2678:dim
2676:and
2330:dim(
2115:are
2085:Let
1634:) =
1584:unit
1570:for
1534:) =
1520:via
1420:and
1405:dim(
1383:rank
1365:and
1342:and
1338:The
1323:) =
1261:(if
957:(⋅,
953:) =
938:, ⋅)
930:) =
900:) =
869:) =
381:The
341:Let
299:ring
257:The
240:) =
221:and
204:) +
192:) =
160:) =
141:and
124:) +
112:) =
23:, a
4233:Zbl
4127:Zbl
4038:Zbl
3827:of
3799:↦ (
3764:↦ (
3650:),
3577:on
3368:if
3286:or
3278:= 4
3069:or
2990:to
2980:to
2922:≠ 2
2910:of
2889:of
2838:of
2807:to
2473:is
2362:is
2328:is
2318:is
2311:V/W
2185:= 0
2122:if
2111:in
2075:in
2037:if
1838:).
1803:of
1754:in
1715:if
1707:or
1693:in
1672:if
1660:in
1621:if
1562:is
1558:If
1516:on
1466:if
1462:is
1433:to
1385:of
1377:If
1356:of
1303:of
1287:in
1225:If
1212:to
1173:if
1159:= 0
1085:and
1083:= 0
989:or
973:).
851:by
810:on
478:× 1
459:× 1
261:on
19:In
4877::
4411:–
4363:.
4351:,
4345:,
4321:,
4281:,
4277:,
4254:,
4231:,
4221:,
4213:,
4141:,
4125:,
4115:,
4107:,
4064:,
4036:,
4026:,
4018:,
3833:.
3815:))
3811:,
3803:↦
3791:→
3780:))
3776:,
3768:↦
3756:→
3738:→
3734:×
3722:.
3720:)⟩
3712:,
3704:,
3696:→
3692:×
3673:↦
3665:→
3638:,
3630:→
3626:×
3607:↦
3599:→
3587:.
3582:×
3553:,
3545:→
3541:×
3532:.
3527:∈
3523:,
3513:∈
3509:,
3499:∈
3495:,
3479:,
3473:αB
3469:xβ
3467:,
3465:αu
3451:,
3439:,
3427:+
3423:,
3407:,
3395:,
3383:,
3379:+
3359:→
3355:×
3309:.
3299:,
3234:=
3230:+
3219:,
3058:ij
3047:.
3042:→
3034:xy
3028:,
3017:→
3013:×
3004:.
2968:→
2964:×
2936:.
2924:,
2897:(Λ
2874:.
2869:⊗
2854:⊗
2843:⊗
2829:→
2825:⊗
2817:.
2801:,
2790:⊗
2786:→
2782:×
2768:→
2764:⊗
2747:,
2739:↦
2735:⊗
2713:→
2709:⊗
2645:.
2639:,
2631:↦
2623:→
2609:→
2592:→
2588:×
2498:,
2493:∈
2386:,
2381:∈
2377:,
2340:.
2313:→
2176:Ax
2131:,
2107:,
2098:→
2094:×
2071:,
2061:,
2046:,
2028:→
2024:×
1860:→
1856::
1849:,
1787:+
1783:,
1779:+
1758:;
1748:,
1736:,
1724:,
1681:,
1654:,
1642:,
1630:,
1601:xy
1595:,
1546:).
1542:)(
1530:,
1507:→
1490:=
1475:,
1335:).
1331:,
1319:,
1217:=
1207:=
1199:xy
1193:,
1178:→
908:,
896:)(
877:,
865:)(
846:→
842::
835:,
785:.
782:AS
451:.
416:,
403:=
400:ij
386:×
378:.
309:.
248:,
242:λB
233:,
212:,
200:,
188:+
184:,
168:,
162:λB
156:,
132:,
120:,
108:,
104:+
83:→
79:×
39:→
35:×
4867:.
4789:)
4513:)
4509:/
4505:(
4415:)
4397:e
4390:t
4383:v
4369:.
3952:.
3830:M
3820:M
3813:x
3809:u
3807:(
3805:B
3801:u
3797:x
3793:M
3789:M
3785:T
3778:x
3774:u
3772:(
3770:B
3766:x
3762:u
3758:M
3754:M
3750:S
3745:R
3740:R
3736:M
3732:M
3728:B
3718:x
3716:(
3714:T
3710:u
3706:x
3702:u
3698:R
3694:M
3690:M
3686:B
3681:)
3679:x
3677:(
3675:T
3671:x
3667:M
3663:M
3659:T
3654:⟩
3652:x
3648:u
3646:(
3644:S
3640:x
3636:u
3632:R
3628:M
3624:M
3620:B
3615:)
3613:u
3611:(
3609:S
3605:u
3601:M
3597:M
3593:S
3584:M
3580:M
3565:)
3563:x
3561:(
3559:u
3555:x
3551:u
3547:R
3543:M
3539:M
3529:R
3525:β
3521:α
3515:M
3511:y
3507:x
3501:M
3497:v
3493:u
3485:β
3483:)
3481:x
3477:u
3475:(
3463:(
3461:B
3455:)
3453:y
3449:u
3447:(
3445:B
3441:x
3437:u
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