408:
1945:. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs.
1812:
3144:
1893:
1534:
3167:: given any theorem in such a plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. A simple example is that the statement "two points determine a unique line, the line passing through these points" has the dual statement that "two lines determine a unique point, the
1367:
In all the dualities discussed before, the dual of an object is of the same kind as the object itself. For example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal a close relation between objects of seemingly different
2117:
assignment to its free variables makes it true. Satisfiability and validity are dual because the invalid formulas are precisely those whose negations are satisfiable, and the unsatisfiable formulas are those whose negations are valid. This can be viewed as a special case of the previous item, with
1836:
of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each
3588:
1889:: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure.
2476:
In other cases, the bidual is not identical with the primal, though there is often a close connection. For example, the dual cone of the dual cone of a set contains the primal set (it is the smallest cone containing the primal set), and is equal if and only if the primal set is a cone.
1916:, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to
2473:), and duality is an involution. In this case the bidual is not usually distinguished, and instead one only refers to the primal and dual. For example, the dual poset of the dual poset is exactly the original poset, since the converse relation is defined by an involution.
4022:(PDE): instead of solving a PDE directly, it may be easier to first solve the PDE in the "weak sense", i.e., find a distribution that satisfies the PDE and, second, to show that the solution must, in fact, be a function. All the standard spaces of distributions —
1920:
on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from
1819:
There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the
4330:
In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of
3741:
5358:
5252:
2825:
3461:
4445:. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed. Therefore, any duality between categories
1334:
are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a
5808:
With increasing level of generality, it turns out, an increasing amount of technical background is helpful or necessary to understand these theorems: the modern formulation of these dualities can be done using
3938:
2490:, known as the "canonical evaluation map". For finite-dimensional vector spaces this is an isomorphism, but these are not identical spaces: they are different sets. In category theory, this is generalized by
119:
has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
5706:
4675:
to emphasize the duality aspect). Therefore, in some cases, proofs of certain statements can be halved, using such a duality phenomenon. Further notions displaying related by such a categorical duality are
5734:, then the dual of the complex (a higher-dimensional generalization of the planar graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the
383:
is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The
4105:
4213:
5469:
3368:
4150:
2935:
554:
4305:
3163:
that map each point of the projective plane to a line, and each line of the projective plane to a point, in an incidence-preserving way. For such planes there arises a general principle of
666:
734:, as illustrated in the diagram. Unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set
5894:
3655:
3243:
1949:
is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph.
4058:
3036:
3453:
3325:
3846:
990:
5257:
5148:
2723:
2622:
3620:
3208:
1988:
1232:
510:
480:
1191:
5406:
3151:, a configuration of four points and six lines in the projective plane (left) and its dual configuration, the complete quadrilateral, with four lines and six points (right).
3806:
2587:
712:
600:
4239:
950:
3088:. Their purpose is to measure angles and distances. Thus, duality is a foundational basis of this branch of geometry. Another application of inner product spaces is the
1580:(short for partially ordered set; i.e., a set that has a notion of ordering but in which two elements cannot necessarily be placed in order relative to each other), the
3663:
1996:; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a
2554:
6243:
3403:
896:
782:
1127:
1080:
3178:
A conceptual explanation of this phenomenon in some planes (notably field planes) is offered by the dual vector space. In fact, the points in the projective plane
5123:; finite groups correspond to finite groups. On the one hand, Pontryagin is a special case of Gelfand duality. On the other hand, it is the conceptual reason of
6270:
3781:
3283:
3263:
1100:
1053:
1033:
1013:
916:
866:
846:
822:
802:
752:
732:
451:
219:
The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case.
3763:
vector space. There are several notions of topological dual space, and each of them gives rise to a certain concept of duality. A topological vector space
6809:
3871:
3406:
1824:, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The
6774:
3583:{\displaystyle \langle \cdot ,\cdot \rangle :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ,\langle x,y\rangle =\sum _{i=1}^{3}x_{i}y_{i}}
5479:. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups
4982:
sets, i.e., there is some notion of an object "being smaller" than another one. A duality that respects the orderings in question is known as a
4175:
4473:). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.
2298:
has operators denoting "will be true at some time in the future" and "will be true at all times in the future" which are similarly dual.
2000:
with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.
5663:
2889:
6617:
4262:
1685:. In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if
7136:
605:
4966:
draws inspiration from
Gelfand duality and studies noncommutative C*-algebras as if they were functions on some imagined space.
1799:
theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the
6651:
4794:
in a natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction
77:. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called
7344:
7237:
7217:
7150:
7097:
7064:
7034:
6982:
6944:
6917:
6884:
6855:
6419:
3003:
4491:
notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example,
1877:-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the
5029:
3172:
1521:
391:
of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set
5495:. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of
6618:"Duality in Mathematics and Physics lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB)"
5411:
3330:
4063:
2242:
is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example in
7303:
517:
4990:
mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension
6011:
1715:; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a
4110:
4018:
are linear functionals on appropriate spaces of functions. They are an important technical means in the theory of
3817:
1259:
7370:
6839:
6664:"A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry"
5868:
3625:
3213:
2095:(but the converse does not hold constructively). From this fundamental logical duality follow several others:
6642:
5817:(with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the
4019:
3744:
3164:
2646:
1205:
is defined. The three properties of the dual cone carry over to this type of duality by replacing subsets of
104:
3416:
3288:
2153:
are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem.
5511:
3945:
3760:
2502:
to the double dual functor. For vector spaces (considered algebraically), this is always an injection; see
1356:
6786:
2510:. There is still a canonical evaluation map, but it is not always injective; if it is, this is known as a
955:
131:
from an object of one type and another object of the second type to some family of scalars. For instance,
6825:
6637:
6104:
3956:
3949:
1654:
4025:
2951:
mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the
7336:
7209:
6868:
6847:
5814:
4015:
3593:
3181:
2480:
An important case is for vector spaces, where there is a map from the primal space to the double dual,
1953:
168:
138:
6328:
More generally, one can consider the projective planes over any field, such as the complex numbers or
3413:, this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to
2503:
1964:
1208:
486:
456:
7389:
6930:
4978:
In a number of situations, the two categories which are dual to each other are actually arising from
4967:
4425:
3160:
1164:
167:, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the
82:
17:
6746:
6139:, This quote is the first sentence of the final section named comments in this single-paged-document
5382:
3265:
of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional
1668:
4944:
are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence
3756:
2937:
mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e.,
2518:
1886:
671:
559:
283:
58:
4900:. Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way.
4222:
3736:{\displaystyle \left\{w\in \mathbb {R} ^{3},\langle v,w\rangle =0{\text{ for all }}v\in V\right\}}
2959:
is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.
929:
5962:
5786:
5517:
4963:
4630:
2562:, with different possible topologies on the dual, each of which defines a different bidual space
1815:
The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed.
246:
5139:, problems are frequently solved by passing to the dual description of functions and operators.
2641:
In other cases, showing a relation between the primal and bidual is a significant result, as in
1764:: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are
6741:
6070:
6016:
6001:
5767:
5739:
5100:. Pontryagin duality states that the character group is again locally compact abelian and that
4340:
4320:
3168:
2495:
1960:
may be specified by a system of real variables (the coordinates for a point in
Euclidean space
1930:
1922:
1765:
1581:
1561:
1336:
146:
corresponds to the pairing in which one integrates a distribution against a test function, and
50:
31:
2596:
1471:
of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction:
6663:
6632:
6409:
5726:
Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological
5533:
5136:
5097:
4336:
3124:
2526:
1993:
1661:
94:
6218:
5353:{\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi .}
3373:
871:
757:
7354:
7324:
7283:
7247:
7193:
7107:
7074:
7044:
6992:
6954:
6894:
6876:
6821:
6805:
6712:
6697:
6085:
5991:
5897:
5525:
5365:
5247:{\displaystyle {\widehat {f}}(\xi ):=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,}
5033:
5022:
4953:
4668:
2997:
is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate
2820:{\displaystyle X\to X^{**}:=(X^{*})^{*}=\operatorname {Hom} (\operatorname {Hom} (X,D),D).}
1781:
1777:
1704:
1105:
1058:
8:
6346:
6045:
5980:
4957:
4685:
4308:
3786:
3658:
3148:
3067:
3045:
2567:
2522:
2154:
1198:
384:
153:
108:
5604:
2534:
1238:
Applying the operation of taking the dual vector space twice gives another vector space
148:
6255:
6090:
5958:
5954:
5763:
5654:
5048:
4920:
4904:
4311:
with values in the circle (with multiplication of complex numbers as group operation).
3766:
3268:
3248:
3127:. In this guise, the duality inherent in the inner product space exchanges the role of
2642:
2511:
2040:
1085:
1038:
1018:
998:
901:
851:
831:
807:
787:
737:
717:
436:
388:
54:
7254:
Negrepontis, Joan W. (1971), "Duality in analysis from the point of view of triples",
4956:. The previous result therefore tells that the local theory of schemes is the same as
7366:
7340:
7299:
7271:
7267:
7233:
7213:
7181:
7146:
7132:
7093:
7060:
7030:
6978:
6958:
6940:
6913:
6880:
6864:
6851:
6798:
6762:
6685:
6415:
6075:
5841:
5771:
5650:
5589:
5554:
5503:
5496:
5142:
4983:
4979:
4941:
4677:
4492:
4429:
4253:
4216:
3103:
3085:
3063:
2631:
1846:
1833:
1708:
1596:
1159:
1150:
135:
corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the
128:
7202:
5818:
7263:
7171:
7119:
7022:
6926:
6901:
6751:
6675:
6628:
6021:
5845:
5810:
5802:
5798:
5782:
5619:
5608:
5124:
4908:
4681:
4428:. There are various situations, where such a functor is an equivalence between the
3156:
3093:
2499:
2243:
2036:
1373:
7007:
6680:
281:. This is referred to by saying that the operation of taking the complement is an
7350:
7320:
7279:
7243:
7189:
7142:
7103:
7089:
7070:
7056:
7040:
7018:
6988:
6974:
6950:
6936:
6909:
6890:
6817:
6701:(a non-technical overview about several aspects of geometry, including dualities)
6693:
6026:
5975:
5790:
5581:
5507:
5063:
5052:
5010:
4931:
4866:
4852:
4735:
4488:
4332:
4250:
4246:
4166:
3809:
2952:
2635:
1946:
1938:
1917:
1854:
1850:
1838:
1825:
1755:
1747:
160:
5821:
in the second case, and with respect to coherent sheaves for coherent duality).
2874:. Depending on the concrete duality considered and also depending on the object
6725:
6080:
6065:
6060:
5623:
5549:(in the sense of linear algebra) of other objects of interest are often called
5476:
5116:
4697:
4664:
4511:
3410:
3132:
3128:
3059:
2657:
A group of dualities can be described by endowing, for any mathematical object
2627:
2307:
2295:
2101:
1957:
1842:
1821:
1795:
are dual concepts: the complement of an open set is closed, and vice versa. In
1139:
339:
208:
6999:
6756:
4629:. This is a particular case of a more general duality phenomenon, under which
7383:
7312:
7275:
7185:
7124:
6770:
6729:
6689:
6333:
5794:
5120:
5055:
5037:
5014:
4987:
4912:
4863:
4656:
4481:
4242:
3862:
3855:
3077:
2998:
2106:
1537:
1369:
1348:
142:
7227:
6962:
6732:(2008), "The concept of duality for measure projections of convex bodies",
6329:
6031:
5849:
5837:
5829:
5731:
5090:
4652:
4162:
2682:
1997:
1909:
1897:
1878:
1800:
1386:
1143:
4219:
containing the lattice that map the points of the lattice to the integers
1552:, is obtained by turning the diagram upside-down. The green nodes form an
7295:
7159:
7003:
6055:
6041:
5985:
5957:
of finite groups. For local and global fields, similar statements exist (
5833:
5825:
5472:
5018:
4693:
4648:
3055:
2507:
2217:
1829:
1263:
407:
38:
7176:
5471:. Moreover, the transform interchanges operations of multiplication and
2469:, depending on context, is often identical to the original (also called
1234:
by vector space and inclusions of such subsets by linear maps. That is:
7081:
7026:
6249:
6246:
5996:
5751:
5546:
4856:
3245:
while the lines in the projective plane correspond to subvector spaces
3089:
2938:
2626:
is not in general an isomorphism. If it is, this is known (for certain
1913:
1901:
1811:
1792:
828:
Applying the operation twice gives back a possibly bigger set: for all
364:
123:
Many mathematical dualities between objects of two types correspond to
2685:. In general, this yields a true duality only for specific choices of
6100:
6095:
6036:
6006:
4689:
4672:
4660:
3960:
3941:
1773:
1769:
1716:
1553:
430:
2962:
5905:
5727:
5529:
5521:
4893:
3933:{\displaystyle H\to H^{*},v\mapsto (w\mapsto \langle w,v\rangle ),}
3081:
1992:
a system of linear constraints (specifying that the point lie in a
1832:
of the center points of each face of the primal polyhedron, so the
1788:
1631:
426:
360:
356:
5824:
Yet another group of similar duality statements is encountered in
5801:
are similar to the statements above, but applies to cohomology of
4876:
the space of continuous functions (which vanish at infinity) from
4704:
4638:
3143:
1912:, the graph of its vertices and edges. The dual polyhedron has a
1796:
1556:
and a lower set in the original and the dual order, respectively.
164:
124:
46:
5852:
of the field) admit similar pairings. The absolute Galois group
1456:
Compared to the above, this duality has the following features:
156:, viewed as a pairing between submanifolds of a given manifold.
5781:-coefficients instead. This is further generalized to possibly
5757:
5701:{\displaystyle (\gamma ,\omega )\mapsto \int _{\gamma }\omega }
3750:
2039:. The basic duality of this type is the duality of the ∃ and ∀
1892:
824:. Therefore this duality is weaker than the one above, in that
228:
1615:. This gives rise to the first example of a duality mentioned
1611:
on any collection of sets, such as the subsets of a fixed set
5499:
systems in terms of coordinate and momentum representations.
4998:(inside some fixed bigger field Ω) the Galois group Gal (Ω /
1881:
of the primal and dual polyhedra or polytopes are themselves
1567:
7232:(2nd ed.), Charleston, South Carolina: BookSurge, LLC,
3092:
which provides a correspondence between the elements of the
2294:. Other dual modal operators behave similarly. For example,
1885:. Duality of polytopes and order-theoretic duality are both
5510:
of multiplication by polynomials with constant coefficient
5145:
switches between functions on a vector space and its dual:
4810:
between the colimit functor that assigns to any diagram in
3759:, a similar construction exists, replacing the dual by the
3590:
yields an identification of this projective plane with the
1368:
nature. One example of such a more general duality is from
215:
Duality in mathematics is not a theorem, but a "principle".
6779:
Electronic
Research Announcements in Mathematical Sciences
5649:. Poincaré duality can also be expressed as a relation of
5545:
Theorems showing that certain objects of interest are the
5005:
The collection of all open subsets of a topological space
4818:
its colimit and the diagonal functor that maps any object
2514:; if it is an isomophism, the module is called reflexive.
387:
of a set is the largest open set contained in it, and the
4325:
5904:, the integers. Therefore, the perfect pairing (for any
5520:
is an important analytic duality which switches between
4930:, one gets back a ring by taking global sections of the
3102:-dimensional vector space, the Hodge star operator maps
2306:
Set-theoretic union and intersection are dual under the
1533:
7008:"An introduction to Tannaka duality and quantum groups"
5844:, since étale cohomology over a field is equivalent to
952:
is turned into an inclusion in the opposite direction (
7162:(1973), "Notes on étale cohomology of number fields",
4480:. An example of self-dual category is the category of
1908:
From any three-dimensional polyhedron, one can form a
1520:
are inverse to each other. This is the content of the
266:. Taking the complement has the following properties:
6768:
6724:
6502:, Theorem VI.1.1) for finite Galois extensions.
6407:
6292:
6281:
6258:
6221:
5871:
5666:
5414:
5385:
5260:
5151:
4970:
is a non-commutative analogue of
Pontryagin duality.
4265:
4225:
4208:{\displaystyle \operatorname {Hom} (L,\mathbb {Z} ),}
4178:
4113:
4066:
4028:
3874:
3820:
3789:
3769:
3666:
3628:
3596:
3464:
3419:
3376:
3333:
3291:
3271:
3251:
3216:
3184:
3006:
2892:
2726:
2599:
2570:
2537:
1967:
1211:
1167:
1108:
1088:
1061:
1041:
1021:
1001:
958:
932:
904:
874:
854:
834:
810:
790:
760:
740:
720:
674:
608:
562:
520:
489:
459:
439:
270:
Applying it twice gives back the original set, i.e.,
4476:
A category that is equivalent to its dual is called
3066:
bilinear form gives rise to such an isomorphism. In
922:
The other two properties carry over without change:
7164:
5865:) of a finite field, for example, is isomorphic to
2105:in a certain model if there are assignments to its
1599:. Familiar examples of dual partial orders include
7201:
6264:
6237:
5888:
5723:-(real) -dimensional cycle) is a perfect pairing.
5700:
5614:is given by a pairing of singular cohomology with
5464:{\displaystyle f(-x)={\widehat {\widehat {f}}}(x)}
5463:
5400:
5352:
5246:
4299:
4233:
4207:
4144:
4099:
4052:
3932:
3840:
3800:
3775:
3735:
3649:
3614:
3582:
3447:
3397:
3363:{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }
3362:
3319:
3277:
3257:
3237:
3202:
3030:
2929:
2819:
2616:
2581:
2548:
2003:
1982:
1226:
1185:
1121:
1094:
1074:
1047:
1027:
1007:
984:
944:
910:
890:
860:
840:
816:
796:
776:
746:
726:
706:
660:
594:
548:
504:
474:
445:
6401:
5506:is similar to Fourier transform and interchanges
4529:of sets are dual to each other in the sense that
4100:{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}
3138:
2941:, forms a vector space in its own right. The map
2250:is possibly true" means "there exists some world
53:into other concepts, theorems or structures in a
7381:
7088:, Graduate Texts in Mathematics, vol. 211,
7055:, Graduate Texts in Mathematics, vol. 189,
7017:, Lecture Notes in Mathematics, vol. 1488,
6705:
4952:Affine schemes are the local building blocks of
2504:Dual space § Injection into the double-dual
1837:other, so do the corresponding two parts of the
1806:
1449:consisting of elements fixed by the elements in
1418:). This group is a subgroup of the Galois group
1138:A very important example of a duality arises in
918:, namely the cones, the two are actually equal.)
6863:
6525:
4453:is formally the same as an equivalence between
3210:correspond to one-dimensional subvector spaces
3171:of these two lines". For further examples, see
2930:{\displaystyle V^{*}=\operatorname {Hom} (V,K)}
2881:
1828:of any of these polyhedra may be formed as the
1279:gives rise to a map in the opposite direction (
549:{\displaystyle C^{*}\subseteq \mathbb {R} ^{2}}
57:fashion, often (but not always) by means of an
30:For the property of optimization problems, see
6908:, Lecture Notes in Mathematics, vol. 20,
6804:
6775:"A characterization of the concept of duality"
6656:, Princeton University Press, pp. 187–190
6448:
6052:, and is unrelated to the notions given above.
4300:{\displaystyle \operatorname {Hom} (G,S^{1}),}
2591:. In these cases the canonical evaluation map
7116:An introduction to abstract harmonic analysis
6816:, Amsterdam: North-Holland, pp. 73–126,
6668:Bulletin of the American Mathematical Society
6411:Locally Presentable and Accessible Categories
5085:given by continuous group homomorphisms from
4884:, the complex numbers. Conversely, the space
3783:that is canonically isomorphic to its bidual
2878:, this map may or may not be an isomorphism.
2506:. This can be generalized algebraically to a
2302:Other analogous dualities follow from these:
2118:the quantifiers ranging over interpretations.
1324:A particular feature of this duality is that
395:is equal to the closure of the complement of
7363:Functional analysis. Theory and applications
5815:direct and inverse image functors of sheaves
5762:The same duality pattern holds for a smooth
5758:Duality in algebraic and arithmetic geometry
4145:{\displaystyle {\mathcal {C}}^{\infty }(U)'}
3921:
3909:
3751:Topological vector spaces and Hilbert spaces
3705:
3693:
3533:
3521:
3477:
3465:
3084:and such positive bilinear forms are called
1882:
1528:
27:General concept and operation in mathematics
7311:
7253:
6998:
6711:James C. Becker and Daniel Henry Gottlieb,
6486:
6460:
6316:
6303:
3948:. As a corollary, every Hilbert space is a
3854:As in the finite-dimensional case, on each
2286:then follows from the analogous duality of
2266:is necessarily true" means "for all worlds
2043:in classical logic. These are dual because
1952:A kind of geometric duality also occurs in
1616:
661:{\displaystyle x_{1}c_{1}+x_{2}c_{2}\geq 0}
379:is closed if and only if its complement in
6925:
6900:
6713:A History of Duality in Algebraic Topology
6650:Gowers, Timothy (2008), "III.19 Duality",
6564:
6473:
6414:. Cambridge University Press. p. 62.
6048:; the term "dual" here is synonymous with
5540:
2886:The construction of the dual vector space
2129:operators are dual in this sense, because
1956:, but not one that reverses dimensions. A
1865:-dimensional polytope corresponding to an
7319:, Blaisdell Publishing Co. Ginn and Co.,
7175:
7123:
6755:
6745:
6679:
6627:
6136:
5889:{\displaystyle {\widehat {\mathbf {Z} }}}
5584:, there is, therefore, an isomorphism of
5553:. Many of these dualities are given by a
5340:
5234:
4227:
4195:
4084:
3680:
3650:{\displaystyle V\subset \mathbb {R} ^{3}}
3637:
3602:
3599:
3514:
3500:
3485:
3425:
3356:
3342:
3297:
3238:{\displaystyle V\subset \mathbb {R} ^{3}}
3225:
3190:
3187:
1970:
1214:
536:
492:
462:
227:A simple duality arises from considering
222:
163:viewpoint, duality can also be seen as a
7365:. New York: Holt, Rinehart and Winston.
7138:Categories for the Working Mathematician
7131:
6810:"Homotopy theories and model categories"
6382:
4986:. An example is the standard duality in
4903:In a similar vein there is a duality in
4647:; further concrete examples of this are
3142:
2720:, that is to say, the dual of the dual,
2649:is naturally isomorphic to its bidual).
2184:. The left side is true if and only if
1891:
1810:
1532:
406:
202:
7360:
6968:
6661:
6551:
6358:
4847:
4156:
3747:arise by means of this identification.
3031:{\displaystyle \varphi :V\times V\to K}
2558:to distinguish from the algebraic dual
2079:in classical logic: if there exists an
1841:. More generally, using the concept of
14:
7382:
7333:An introduction to homological algebra
7330:
7113:
6838:
6653:The Princeton Companion to Mathematics
6649:
6615:
6512:
6435:
6371:
6172:
6160:
6154:
6148:
6142:
6124:
4948:(Commutative rings) ≅ (affine schemes)
4326:Opposite category and adjoint functors
4241:. This is used in the construction of
4152:— are reflexive locally convex spaces.
3458:The (positive definite) bilinear form
3448:{\displaystyle (\mathbb {R} ^{3})^{*}}
3320:{\displaystyle (\mathbb {R} ^{3})^{*}}
3123:-forms. This can be used to formulate
2677:, with a structure similar to that of
2200:, and the right side if and only if ¬∃
1732:A concept defined for a partial order
1703:are two duality transforms then their
1595:comprises the same ground set but the
7289:
7225:
7199:
7158:
6718:
6609:
6594:
6590:
6577:
6538:
6203:
5130:
5043:
4973:
4926:. Conversely, given an affine scheme
3622:. Concretely, the duality assigns to
2087:fails to hold, then it is false that
7080:
6875:, Wiley Classics Library, New York:
6524:Griffiths & Harris
6499:
1522:fundamental theorem of Galois theory
1133:
985:{\displaystyle D^{*}\subseteq C^{*}}
514:the dual cone is defined as the set
7292:Principles Of Mathematical Analysis
7118:, D. Van Nostrand, pp. x+190,
7050:
6476:, Ch. II.2, esp. Prop. II.2.3
6395:
5599:
5051:gives a duality on the category of
4215:the set of linear functions on the
2837:the map that associates to any map
2525:), there is a separate notion of a
2365:. This follows from the duality of
1929:of points in the plane between the
1548:. The dual poset, i.e. ordering by
1433:. Conversely, to any such subgroup
1343:. This is also true in the case if
926:It is still true that an inclusion
300:is turned into an inclusion in the
171:construction assigns to each arrow
81:). Such involutions sometimes have
24:
6447:Dwyer and Spaliński (
6184:The complement is also denoted as
5289:
5284:
5189:
5184:
4960:, the study of commutative rings.
4826:to the constant diagram which has
4424:That functor may or may not be an
4314:
4123:
4117:
4070:
4053:{\displaystyle {\mathcal {D}}'(U)}
4032:
2075:are equivalent for all predicates
1603:the subset and superset relations
254:consists of all those elements in
25:
7401:
6315:(Veblen & Young
6293:Artstein-Avidan & Milman 2008
6282:Artstein-Avidan & Milman 2007
5030:Birkhoff's representation theorem
4855:is a duality between commutative
3615:{\displaystyle \mathbb {RP} ^{2}}
3203:{\displaystyle \mathbb {RP} ^{2}}
2491:
2461:The dual of the dual, called the
2008:In logic, functions or relations
1925:: the duality for any finite set
6873:Principles of algebraic geometry
6408:Jiří Adámek; J. Rosicky (1994).
6012:Duality (electrical engineering)
5876:
3327:consisting of those linear maps
1983:{\displaystyle \mathbb {R} ^{n}}
1362:
1260:finite-dimensional vector spaces
1227:{\displaystyle \mathbb {R} ^{2}}
784:is the smallest cone containing
505:{\displaystyle \mathbb {R} ^{n}}
475:{\displaystyle \mathbb {R} ^{2}}
367:of some fixed topological space
7317:Projective geometry. Vols. 1, 2
6844:Introduction to toric varieties
6583:
6570:
6557:
6544:
6531:
6518:
6505:
6492:
6485:Joyal and Street (
6479:
6466:
6454:
6441:
6428:
6388:
6376:
6364:
6352:
6339:
6322:
6309:
6297:
6286:
5013:. There is a duality, known as
4696:in topology and more generally
2652:
2157:are examples. More generally,
2004:Duality in logic and set theory
1725:are induced by permutations of
1645:relations on the set of humans.
1186:{\displaystyle \varphi :V\to K}
6814:Handbook of algebraic topology
6734:Journal of Functional Analysis
6275:
6209:
6197:
6178:
6166:
6130:
6118:
5682:
5679:
5667:
5645:is the (complex) dimension of
5458:
5452:
5427:
5418:
5401:{\displaystyle {\widehat {f}}}
5312:
5306:
5270:
5264:
5203:
5197:
5170:
5164:
4291:
4272:
4199:
4185:
4135:
4128:
4094:
4079:
4047:
4041:
4020:partial differential equations
3924:
3906:
3900:
3897:
3878:
3745:duality in projective geometry
3510:
3436:
3420:
3386:
3380:
3352:
3308:
3292:
3139:Duality in projective geometry
3022:
2924:
2912:
2811:
2802:
2790:
2781:
2763:
2749:
2730:
2603:
1177:
701:
675:
589:
563:
13:
1:
7315:; Young, John Wesley (1965),
7053:Lectures on modules and rings
6706:Duality in algebraic topology
6681:10.1090/S0273-0979-01-00913-2
6604:
5618:-coefficients (equivalently,
5512:linear differential operators
4919:there is an affine spectrum,
2712:. There is always a map from
2647:locally compact abelian group
1807:Dimension-reversing dualities
707:{\displaystyle (c_{1},c_{2})}
595:{\displaystyle (x_{1},x_{2})}
482:(or more generally points in
7268:10.1016/0021-8693(71)90105-0
6515:, p. 151, section 37D)
5711:(integrating a differential
5657:, by asserting that the map
4915:: to every commutative ring
4234:{\displaystyle \mathbb {Z} }
3946:Riesz representation theorem
3165:duality in projective planes
2882:Dual vector spaces revisited
1357:Riesz representation theorem
1310:correspond to the maps from
945:{\displaystyle C\subseteq D}
402:
7:
7331:Weibel, Charles A. (1994),
7229:Arithmetic duality theorems
6638:Encyclopedia of Mathematics
5968:
5953:is a direct consequence of
4872:is the same: it assigns to
3841:{\displaystyle X\cong X''.}
3743:. The explicit formulas in
2681:. This is sometimes called
2231:is "necessarily" true, and
2227:means that the proposition
2109:that render it true; it is
1861:-dimensional feature of an
1655:involutive antiautomorphism
556:consisting of those points
10:
7406:
7337:Cambridge University Press
7294:(3rd ed.), New York:
7210:Princeton University Press
6848:Princeton University Press
6319:, Ch. I, Theorem 11)
6163:, p. 189, col. 2
6151:, p. 187, col. 1
6071:Linear programming#Duality
5789:instead, a duality called
4888:can be reconstructed from
4368:which for any two objects
4318:
3405:. As a consequence of the
1857:or dual polytope, with an
1559:
1400:to any intermediate field
433:construction. Given a set
262:. It is again a subset of
258:that are not contained in
115:In mathematical contexts,
61:operation: if the dual of
29:
6808:; Spaliński, Jan (1995),
6757:10.1016/j.jfa.2007.11.008
5963:Poitou–Tate duality
4814:indexed by some category
4641:in the opposite category
4426:equivalence of categories
3757:topological vector spaces
3161:geometric transformations
3159:, it is possible to find
2519:topological vector spaces
2456:
1529:Order-reversing dualities
1443:there is the fixed field
804:which may be bigger than
415:(blue) and its dual cone
152:corresponds similarly to
7226:Milne, James S. (2006),
7200:Milne, James S. (1980),
7114:Loomis, Lynn H. (1953),
6969:Iversen, Birger (1986),
6769:Artstein-Avidan, Shiri;
6662:Cartier, Pierre (2001),
6616:Atiyah, Michael (2007).
6554:, Ch. VII.3, VII.5
6111:
5487:etc.): any character of
5096:can be endowed with the
2975:and inner product spaces
2617:{\displaystyle V\to V''}
2121:In classical logic, the
2099:A formula is said to be
1849:, or more generally any
1385:, one may associate the
1292:Given two vector spaces
1244:. There is always a map
355:This duality appears in
101:in this sense under the
7361:Edwards, R. E. (1965).
7051:Lam, Tsit-Yuen (1999),
6785:: 42–59, archived from
6580:, Example I.1.10)
6304:Veblen & Young 1965
5787:intersection cohomology
5541:Homology and cohomology
5518:Legendre transformation
5058:: given any such group
5002:) —to a smaller group.
4964:Noncommutative geometry
4830:at all places. Dually,
2673:into some fixed object
2661:, the set of morphisms
2630:vector spaces with the
2016:are considered dual if
1544:, partially ordered by
1258:, namely precisely the
1158:. Its elements are the
453:of points in the plane
51:mathematical structures
7290:Rudin, Walter (1976),
6726:Artstein-Avidan, Shiri
6266:
6239:
6238:{\displaystyle C^{**}}
6017:Duality (optimization)
6002:Dual (category theory)
5890:
5828:: étale cohomology of
5768:separably closed field
5702:
5465:
5402:
5354:
5248:
5121:compact abelian groups
4321:Dual (category theory)
4301:
4235:
4209:
4146:
4101:
4054:
3950:reflexive Banach space
3934:
3842:
3802:
3777:
3737:
3651:
3616:
3584:
3559:
3449:
3399:
3398:{\displaystyle f(V)=0}
3364:
3321:
3279:
3259:
3239:
3204:
3152:
3032:
2931:
2821:
2704:is referred to as the
2618:
2583:
2550:
2496:natural transformation
2333:, and more generally,
1984:
1931:Delaunay triangulation
1923:computational geometry
1905:
1816:
1766:upper and lower bounds
1562:Duality (order theory)
1557:
1228:
1187:
1142:by associating to any
1123:
1096:
1076:
1049:
1029:
1009:
986:
946:
912:
892:
891:{\displaystyle C^{**}}
862:
842:
818:
798:
778:
777:{\displaystyle C^{**}}
748:
728:
708:
662:
596:
550:
506:
476:
447:
422:
223:Complement of a subset
217:
133:linear algebra duality
85:, so that the dual of
32:Duality (optimization)
6971:Cohomology of sheaves
6877:John Wiley & Sons
6267:
6240:
5891:
5703:
5534:Hamiltonian mechanics
5475:on the corresponding
5466:
5403:
5355:
5249:
5098:compact-open topology
5034:distributive lattices
4968:Tannaka–Krein duality
4659:(or groups etc.) vs.
4337:contravariant functor
4302:
4236:
4210:
4147:
4102:
4055:
3935:
3843:
3803:
3778:
3738:
3652:
3617:
3585:
3539:
3450:
3400:
3365:
3322:
3280:
3260:
3240:
3205:
3146:
3033:
2932:
2851:(i.e., an element in
2822:
2619:
2584:
2551:
2416:, and is a member of
1985:
1895:
1883:order-theoretic duals
1814:
1736:will correspond to a
1662:partially ordered set
1536:
1229:
1188:
1124:
1122:{\displaystyle C^{*}}
1097:
1077:
1075:{\displaystyle D^{*}}
1050:
1030:
1010:
987:
947:
913:
893:
863:
843:
819:
799:
779:
749:
729:
709:
663:
597:
551:
507:
477:
448:
410:
359:as a duality between
290:An inclusion of sets
213:
203:Introductory examples
93:itself. For example,
45:translates concepts,
7021:, pp. 413–492,
6906:Residues and Duality
6256:
6219:
5992:Dual abelian variety
5898:profinite completion
5869:
5730:is represented as a
5664:
5607:of a smooth compact
5526:Lagrangian mechanics
5412:
5383:
5258:
5149:
4848:Spaces and functions
4335:, this amounts to a
4263:
4223:
4176:
4157:Further dual objects
4111:
4064:
4026:
3872:
3818:
3787:
3767:
3664:
3626:
3594:
3462:
3417:
3374:
3331:
3289:
3269:
3249:
3214:
3182:
3004:
2890:
2724:
2597:
2568:
2535:
2523:normed vector spaces
1965:
1540:of the power set of
1209:
1165:
1106:
1086:
1059:
1039:
1019:
999:
956:
930:
902:
872:
852:
832:
808:
788:
758:
738:
718:
672:
606:
560:
518:
487:
457:
437:
7177:10.24033/asens.1257
6347:elliptic regularity
6046:associative algebra
5981:Autonomous category
5293:
5193:
4958:commutative algebra
4738:if for all objects
4686:homological algebra
4309:group homomorphisms
3801:{\displaystyle X''}
3716: for all
3149:complete quadrangle
3125:Maxwell's equations
3076:is taken to be the
3068:Riemannian geometry
3046:inner product space
2827:It assigns to some
2582:{\displaystyle V''}
2492:§ Dual objects
2278:". The duality of
1954:optimization theory
1853:, corresponds to a
1843:polar reciprocation
429:is provided by the
154:intersection number
141:and the associated
109:projective geometry
69:, then the dual of
7256:Journal of Algebra
7133:Mac Lane, Saunders
7027:10.1007/BFb0084235
6932:Algebraic Geometry
6912:, pp. 20–48,
6865:Griffiths, Phillip
6719:Specific dualities
6610:Duality in general
6262:
6235:
6091:Pontryagin duality
5955:Pontryagin duality
5886:
5848:of the (absolute)
5811:derived categories
5783:singular varieties
5764:projective variety
5698:
5655:de Rham cohomology
5497:quantum mechanical
5461:
5398:
5379:, say, then so is
5350:
5276:
5244:
5176:
5131:Analytic dualities
5049:Pontryagin duality
5044:Pontryagin duality
4974:Galois connections
4942:ring homomorphisms
4905:algebraic geometry
4493:Cartesian products
4489:category-theoretic
4297:
4254:topological groups
4231:
4205:
4142:
4097:
4050:
3994:, but the dual of
3930:
3838:
3798:
3773:
3733:
3647:
3612:
3580:
3445:
3395:
3360:
3317:
3275:
3255:
3235:
3200:
3169:intersection point
3153:
3086:Riemannian metrics
3048:. For example, if
3028:
2927:
2817:
2643:Pontryagin duality
2614:
2579:
2549:{\displaystyle V'}
2546:
2512:torsionless module
1980:
1906:
1817:
1746:. For instance, a
1740:on the dual poset
1709:order automorphism
1558:
1224:
1183:
1160:linear functionals
1119:
1092:
1072:
1045:
1025:
1005:
995:Given two subsets
982:
942:
908:
888:
858:
838:
814:
794:
774:
744:
724:
704:
658:
592:
546:
502:
472:
443:
423:
317:Given two subsets
129:bilinear functions
95:Desargues' theorem
7346:978-0-521-55987-4
7239:978-1-4196-4274-6
7219:978-0-691-08238-7
7152:978-0-387-98403-2
7125:2027/uc1.b4250788
7099:978-0-387-95385-4
7066:978-0-387-98428-5
7036:978-3-540-46435-8
6984:978-3-540-16389-3
6946:978-0-387-90244-9
6927:Hartshorne, Robin
6919:978-3-540-34794-1
6902:Hartshorne, Robin
6886:978-0-471-05059-9
6857:978-0-691-00049-7
6806:Dwyer, William G.
6567:, Ch. III.7
6541:, Ch. VI.11
6421:978-0-521-42261-1
6265:{\displaystyle C}
6076:List of dualities
5883:
5842:Galois cohomology
5772:l-adic cohomology
5651:singular homology
5504:Laplace transform
5449:
5444:
5395:
5317:
5303:
5208:
5161:
5143:Fourier transform
5009:forms a complete
4984:Galois connection
4980:partially ordered
4909:commutative rings
4682:injective modules
4430:opposite category
4217:real vector space
4012:is not reflexive.
3957:dual normed space
3776:{\displaystyle X}
3717:
3407:dimension formula
3278:{\displaystyle V}
3258:{\displaystyle W}
3157:projective planes
3064:positive definite
2985:is isomorphic to
2632:strong dual space
1900:in blue, and its
1847:convex polyhedron
1651:duality transform
1630:relations on the
1597:converse relation
1262:, this map is an
1151:dual vector space
1134:Dual vector space
1095:{\displaystyle D}
1048:{\displaystyle C}
1028:{\displaystyle D}
1008:{\displaystyle C}
911:{\displaystyle C}
861:{\displaystyle C}
841:{\displaystyle C}
817:{\displaystyle C}
797:{\displaystyle C}
747:{\displaystyle C}
727:{\displaystyle C}
446:{\displaystyle C}
16:(Redirected from
7397:
7390:Duality theories
7376:
7357:
7327:
7308:
7286:
7250:
7222:
7207:
7204:Étale cohomology
7196:
7179:
7155:
7141:(2nd ed.),
7128:
7127:
7110:
7077:
7047:
7012:
6995:
6973:, Universitext,
6965:
6922:
6897:
6860:
6835:
6834:
6833:
6824:, archived from
6796:
6795:
6794:
6760:
6759:
6749:
6700:
6683:
6657:
6645:
6629:Kostrikin, A. I.
6624:
6622:
6598:
6587:
6581:
6574:
6568:
6563:Hartshorne
6561:
6555:
6548:
6542:
6535:
6529:
6522:
6516:
6509:
6503:
6496:
6490:
6483:
6477:
6472:Hartshorne
6470:
6464:
6461:Negrepontis 1971
6458:
6452:
6445:
6439:
6432:
6426:
6425:
6405:
6399:
6392:
6386:
6380:
6374:
6368:
6362:
6356:
6350:
6343:
6337:
6326:
6320:
6313:
6307:
6301:
6295:
6290:
6284:
6279:
6273:
6271:
6269:
6268:
6263:
6252:cone containing
6245:is the smallest
6244:
6242:
6241:
6236:
6234:
6233:
6215:More precisely,
6213:
6207:
6201:
6195:
6193:
6182:
6176:
6170:
6164:
6158:
6152:
6146:
6140:
6134:
6128:
6122:
6022:Dualizing module
5895:
5893:
5892:
5887:
5885:
5884:
5879:
5874:
5846:group cohomology
5803:coherent sheaves
5799:coherent duality
5707:
5705:
5704:
5699:
5694:
5693:
5620:sheaf cohomology
5609:complex manifold
5605:Poincaré duality
5600:Poincaré duality
5582:perfect pairings
5555:bilinear pairing
5491:is given by ξ ↦
5470:
5468:
5467:
5462:
5451:
5450:
5445:
5437:
5435:
5407:
5405:
5404:
5399:
5397:
5396:
5388:
5359:
5357:
5356:
5351:
5339:
5338:
5315:
5305:
5304:
5296:
5292:
5287:
5253:
5251:
5250:
5245:
5233:
5232:
5206:
5192:
5187:
5163:
5162:
5154:
5125:Fourier analysis
4867:Hausdorff spaces
4843:
4829:
4825:
4821:
4817:
4813:
4807:
4790:
4733:
4719:
4698:model categories
4655:, in particular
4646:
4636:
4628:
4622:
4574:
4528:
4509:
4472:
4468:
4462:
4456:
4452:
4448:
4444:
4440:
4436:
4421:
4365:
4349:
4345:
4306:
4304:
4303:
4298:
4290:
4289:
4240:
4238:
4237:
4232:
4230:
4214:
4212:
4211:
4206:
4198:
4171:
4151:
4149:
4148:
4143:
4141:
4127:
4126:
4121:
4120:
4106:
4104:
4103:
4098:
4093:
4092:
4087:
4078:
4074:
4073:
4059:
4057:
4056:
4051:
4040:
4036:
4035:
4011:
4005:
3999:
3993:
3985:
3973:
3965:
3939:
3937:
3936:
3931:
3890:
3889:
3867:
3860:
3847:
3845:
3844:
3839:
3834:
3807:
3805:
3804:
3799:
3797:
3782:
3780:
3779:
3774:
3761:topological dual
3755:In the realm of
3742:
3740:
3739:
3734:
3732:
3728:
3718:
3715:
3689:
3688:
3683:
3656:
3654:
3653:
3648:
3646:
3645:
3640:
3621:
3619:
3618:
3613:
3611:
3610:
3605:
3589:
3587:
3586:
3581:
3579:
3578:
3569:
3568:
3558:
3553:
3517:
3509:
3508:
3503:
3494:
3493:
3488:
3454:
3452:
3451:
3446:
3444:
3443:
3434:
3433:
3428:
3404:
3402:
3401:
3396:
3369:
3367:
3366:
3361:
3359:
3351:
3350:
3345:
3326:
3324:
3323:
3318:
3316:
3315:
3306:
3305:
3300:
3285:the subspace of
3284:
3282:
3281:
3276:
3264:
3262:
3261:
3256:
3244:
3242:
3241:
3236:
3234:
3233:
3228:
3209:
3207:
3206:
3201:
3199:
3198:
3193:
3122:
3108:
3101:
3094:exterior algebra
3075:
3054:is the field of
3053:
3043:
3037:
3035:
3034:
3029:
2996:
2990:
2984:
2974:
2968:
2963:Isomorphisms of
2958:
2950:
2936:
2934:
2933:
2928:
2902:
2901:
2877:
2873:
2862:
2850:
2836:
2826:
2824:
2823:
2818:
2771:
2770:
2761:
2760:
2745:
2744:
2715:
2711:
2703:
2689:, in which case
2688:
2680:
2676:
2672:
2660:
2625:
2623:
2621:
2620:
2615:
2613:
2590:
2588:
2586:
2585:
2580:
2578:
2557:
2555:
2553:
2552:
2547:
2545:
2527:topological dual
2500:identity functor
2489:
2451:
2431:
2421:
2415:
2395:
2394:
2393:
2381:
2376:
2372:
2368:
2364:
2354:
2350:
2349:
2337:
2332:
2313:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2244:Kripke semantics
2241:
2237:
2230:
2226:
2199:
2183:
2174:
2161:
2155:De Morgan's laws
2152:
2140:
2128:
2124:
2074:
2058:
2037:logical negation
2034:
2015:
2011:
1991:
1989:
1987:
1986:
1981:
1979:
1978:
1973:
1944:
1936:
1928:
1918:graph embeddings
1876:
1864:
1860:
1763:
1753:
1745:
1735:
1728:
1724:
1714:
1702:
1693:
1684:
1666:
1659:
1614:
1610:
1606:
1594:
1579:
1551:
1547:
1543:
1519:
1515:
1509:
1505:
1490:
1470:
1452:
1448:
1442:
1432:
1417:
1403:
1399:
1384:
1374:Galois extension
1346:
1342:
1333:
1327:
1319:
1313:
1309:
1303:
1300:, the maps from
1299:
1295:
1288:
1278:
1257:
1253:
1243:
1233:
1231:
1230:
1225:
1223:
1222:
1217:
1204:
1196:
1192:
1190:
1189:
1184:
1157:
1148:
1128:
1126:
1125:
1120:
1118:
1117:
1102:is contained in
1101:
1099:
1098:
1093:
1081:
1079:
1078:
1073:
1071:
1070:
1055:is contained in
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1014:
1012:
1011:
1006:
991:
989:
988:
983:
981:
980:
968:
967:
951:
949:
948:
943:
917:
915:
914:
909:
897:
895:
894:
889:
887:
886:
868:is contained in
867:
865:
864:
859:
847:
845:
844:
839:
823:
821:
820:
815:
803:
801:
800:
795:
783:
781:
780:
775:
773:
772:
753:
751:
750:
745:
733:
731:
730:
725:
713:
711:
710:
705:
700:
699:
687:
686:
667:
665:
664:
659:
651:
650:
641:
640:
628:
627:
618:
617:
601:
599:
598:
593:
588:
587:
575:
574:
555:
553:
552:
547:
545:
544:
539:
530:
529:
513:
511:
509:
508:
503:
501:
500:
495:
481:
479:
478:
473:
471:
470:
465:
452:
450:
449:
444:
420:
414:
398:
394:
382:
378:
374:
370:
350:
345:is contained in
344:
338:
333:is contained in
332:
328:
324:
320:
313:
299:
280:
265:
261:
257:
253:
244:
235:. To any subset
234:
207:In the words of
198:
184:
149:Poincaré duality
137:duality between
92:
88:
76:
72:
68:
64:
21:
7405:
7404:
7400:
7399:
7398:
7396:
7395:
7394:
7380:
7379:
7373:
7347:
7306:
7240:
7220:
7153:
7143:Springer-Verlag
7100:
7090:Springer-Verlag
7067:
7057:Springer-Verlag
7037:
7019:Springer-Verlag
7015:Category theory
7010:
6985:
6975:Springer-Verlag
6947:
6937:Springer-Verlag
6920:
6910:Springer-Verlag
6887:
6858:
6840:Fulton, William
6831:
6829:
6792:
6790:
6747:10.1.1.417.3470
6740:(10): 2648–66,
6721:
6708:
6620:
6612:
6607:
6602:
6601:
6593:); Milne (
6588:
6584:
6575:
6571:
6562:
6558:
6549:
6545:
6536:
6532:
6523:
6519:
6510:
6506:
6498:See (Lang
6497:
6493:
6484:
6480:
6471:
6467:
6459:
6455:
6446:
6442:
6433:
6429:
6422:
6406:
6402:
6393:
6389:
6381:
6377:
6369:
6365:
6357:
6353:
6344:
6340:
6327:
6323:
6314:
6310:
6302:
6298:
6291:
6287:
6280:
6276:
6257:
6254:
6253:
6226:
6222:
6220:
6217:
6216:
6214:
6210:
6202:
6198:
6192:
6188:
6185:
6183:
6179:
6171:
6167:
6159:
6155:
6147:
6143:
6135:
6131:
6123:
6119:
6114:
6109:
6105:Mirror symmetry
6027:Dualizing sheaf
5988:and polar body.
5976:Adjoint functor
5971:
5875:
5873:
5872:
5870:
5867:
5866:
5864:
5840:(also known as
5791:Verdier duality
5780:
5760:
5742:group and the (
5715:-form over an 2
5689:
5685:
5665:
5662:
5661:
5602:
5561:-vector spaces
5543:
5477:function spaces
5436:
5434:
5433:
5413:
5410:
5409:
5387:
5386:
5384:
5381:
5380:
5322:
5318:
5295:
5294:
5288:
5280:
5259:
5256:
5255:
5254:and conversely
5213:
5209:
5188:
5180:
5153:
5152:
5150:
5147:
5146:
5133:
5117:discrete groups
5064:character group
5053:locally compact
5046:
5011:Heyting algebra
4976:
4940:. In addition,
4939:
4932:structure sheaf
4853:Gelfand duality
4850:
4845:
4841:
4837:
4833:
4827:
4823:
4819:
4815:
4811:
4808:
4805:
4801:
4797:
4792:
4788:
4784:
4780:
4776:
4775:
4770:
4766:
4762:
4761:
4756:
4732:
4728:
4724:
4721:
4718:
4714:
4710:
4707:
4665:direct products
4645:
4642:
4634:
4626:
4623:
4620:
4616:
4613:
4609:
4605:
4602:
4598:
4594:
4591:
4587:
4584:
4580:
4575:
4572:
4569:
4565:
4561:
4558:
4554:
4550:
4547:
4543:
4540:
4536:
4532:
4527:
4524:
4520:
4517:
4514:
4512:disjoint unions
4508:
4505:
4501:
4498:
4495:
4470:
4467:
4464:
4461:
4458:
4454:
4450:
4446:
4442:
4438:
4435:
4432:
4422:
4419:
4415:
4411:
4407:
4403:
4402:
4397:
4393:
4389:
4388:
4383:
4366:
4364:
4360:
4356:
4353:
4347:
4343:
4333:category theory
4328:
4323:
4317:
4315:Dual categories
4285:
4281:
4264:
4261:
4260:
4251:locally compact
4247:Pontryagin dual
4243:toric varieties
4226:
4224:
4221:
4220:
4194:
4177:
4174:
4173:
4169:
4159:
4134:
4122:
4116:
4115:
4114:
4112:
4109:
4108:
4088:
4083:
4082:
4069:
4068:
4067:
4065:
4062:
4061:
4031:
4030:
4029:
4027:
4024:
4023:
4010:
4007:
4004:
4001:
4000:is bigger than
3998:
3995:
3991:
3987:
3983:
3979:
3975:
3972:
3969:
3964:
3961:
3885:
3881:
3873:
3870:
3869:
3865:
3858:
3827:
3819:
3816:
3815:
3810:reflexive space
3790:
3788:
3785:
3784:
3768:
3765:
3764:
3753:
3714:
3684:
3679:
3678:
3671:
3667:
3665:
3662:
3661:
3641:
3636:
3635:
3627:
3624:
3623:
3606:
3598:
3597:
3595:
3592:
3591:
3574:
3570:
3564:
3560:
3554:
3543:
3513:
3504:
3499:
3498:
3489:
3484:
3483:
3463:
3460:
3459:
3439:
3435:
3429:
3424:
3423:
3418:
3415:
3414:
3375:
3372:
3371:
3355:
3346:
3341:
3340:
3332:
3329:
3328:
3311:
3307:
3301:
3296:
3295:
3290:
3287:
3286:
3270:
3267:
3266:
3250:
3247:
3246:
3229:
3224:
3223:
3215:
3212:
3211:
3194:
3186:
3185:
3183:
3180:
3179:
3141:
3133:electric fields
3112:
3104:
3097:
3071:
3060:complex numbers
3049:
3039:
3005:
3002:
3001:
2992:
2986:
2980:
2979:A vector space
2977:
2970:
2964:
2956:
2949:
2945:
2942:
2897:
2893:
2891:
2888:
2887:
2884:
2875:
2871:
2867:
2864:
2860:
2856:
2852:
2849:
2845:
2841:
2838:
2835:
2831:
2828:
2766:
2762:
2756:
2752:
2737:
2733:
2725:
2722:
2721:
2713:
2709:
2701:
2697:
2693:
2690:
2686:
2678:
2674:
2670:
2666:
2662:
2658:
2655:
2636:reflexive space
2634:topology) as a
2606:
2598:
2595:
2594:
2592:
2571:
2569:
2566:
2565:
2563:
2561:
2538:
2536:
2533:
2532:
2530:
2488:
2484:
2481:
2459:
2450:
2449:
2445:
2441:
2437:
2433:
2432:if and only if
2429:
2428:
2424:
2419:
2417:
2414:
2413:
2409:
2405:
2401:
2397:
2396:if and only if
2392:
2391:
2387:
2386:
2385:
2384:
2379:
2378:
2377:is a member of
2374:
2370:
2366:
2362:
2361:
2357:
2352:
2348:
2347:
2343:
2342:
2341:
2340:
2335:
2334:
2330:
2326:
2322:
2318:
2315:
2311:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2239:
2236:
2232:
2228:
2225:
2221:
2212:
2198:
2197:
2193:
2189:
2185:
2182:
2181:
2177:
2172:
2170:
2169:
2165:
2159:
2158:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2094:
2090:
2086:
2082:
2078:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2032:
2028:
2024:
2020:
2017:
2013:
2009:
2006:
1974:
1969:
1968:
1966:
1963:
1962:
1961:
1947:Matroid duality
1942:
1939:Voronoi diagram
1934:
1926:
1874:
1870:
1866:
1862:
1858:
1855:dual polyhedron
1851:convex polytope
1839:dual polyhedron
1826:dual polyhedron
1822:Platonic solids
1809:
1762:
1759:
1756:maximal element
1751:
1748:minimal element
1744:
1741:
1733:
1726:
1722:
1719:
1712:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1679:
1675:
1672:
1669:order-reversing
1664:
1657:
1612:
1608:
1604:
1592:
1588:
1585:
1577:
1573:
1570:
1564:
1549:
1545:
1541:
1531:
1517:
1514:
1511:
1507:
1503:
1499:
1495:
1488:
1484:
1481:′) ⊆ Gal(
1480:
1476:
1472:
1468:
1464:
1461:
1450:
1447:
1444:
1441:
1437:
1434:
1430:
1426:
1422:
1419:
1416:
1412:
1408:
1405:
1401:
1397:
1393:
1389:
1383:
1379:
1376:
1365:
1344:
1340:
1332:
1329:
1325:
1318:
1315:
1311:
1308:
1305:
1301:
1297:
1293:
1287:
1283:
1280:
1277:
1273:
1270:
1255:
1252:
1248:
1245:
1242:
1239:
1218:
1213:
1212:
1210:
1207:
1206:
1202:
1194:
1166:
1163:
1162:
1156:
1153:
1146:
1136:
1113:
1109:
1107:
1104:
1103:
1087:
1084:
1083:
1082:if and only if
1066:
1062:
1060:
1057:
1056:
1040:
1037:
1036:
1020:
1017:
1016:
1000:
997:
996:
976:
972:
963:
959:
957:
954:
953:
931:
928:
927:
903:
900:
899:
879:
875:
873:
870:
869:
853:
850:
849:
833:
830:
829:
809:
806:
805:
789:
786:
785:
765:
761:
759:
756:
755:
739:
736:
735:
719:
716:
715:
695:
691:
682:
678:
673:
670:
669:
668:for all points
646:
642:
636:
632:
623:
619:
613:
609:
607:
604:
603:
583:
579:
570:
566:
561:
558:
557:
540:
535:
534:
525:
521:
519:
516:
515:
496:
491:
490:
488:
485:
484:
483:
466:
461:
460:
458:
455:
454:
438:
435:
434:
419:
416:
412:
405:
396:
392:
380:
376:
372:
368:
349:
346:
342:
337:
334:
330:
326:
322:
318:
312:
308:
305:
298:
294:
291:
279:
275:
271:
263:
259:
255:
252:
249:
243:
239:
236:
232:
231:of a fixed set
225:
205:
186:
172:
161:category theory
90:
86:
74:
70:
66:
62:
35:
28:
23:
22:
15:
12:
11:
5:
7403:
7393:
7392:
7378:
7377:
7371:
7358:
7345:
7328:
7313:Veblen, Oswald
7309:
7304:
7287:
7262:(2): 228–253,
7251:
7238:
7223:
7218:
7197:
7170:(4): 521–552,
7156:
7151:
7129:
7111:
7098:
7078:
7065:
7048:
7035:
6996:
6983:
6966:
6945:
6923:
6918:
6898:
6885:
6869:Harris, Joseph
6861:
6856:
6836:
6802:
6771:Milman, Vitali
6766:
6730:Milman, Vitali
6720:
6717:
6716:
6715:
6707:
6704:
6703:
6702:
6674:(4): 389–408,
6670:, New Series,
6659:
6647:
6625:
6611:
6608:
6606:
6603:
6600:
6599:
6582:
6569:
6556:
6543:
6530:
6517:
6504:
6491:
6478:
6465:
6453:
6440:
6427:
6420:
6400:
6387:
6375:
6363:
6351:
6338:
6334:division rings
6321:
6308:
6296:
6285:
6274:
6261:
6232:
6229:
6225:
6208:
6196:
6190:
6186:
6177:
6165:
6153:
6141:
6137:Kostrikin 2001
6129:
6116:
6115:
6113:
6110:
6108:
6107:
6098:
6093:
6088:
6086:Petrie duality
6083:
6081:Matlis duality
6078:
6073:
6068:
6066:Langlands dual
6063:
6061:Koszul duality
6058:
6053:
6039:
6034:
6029:
6024:
6019:
6014:
6009:
6004:
5999:
5994:
5989:
5983:
5978:
5972:
5970:
5967:
5961:and global or
5951:
5950:
5882:
5878:
5860:
5819:étale topology
5778:
5759:
5756:
5709:
5708:
5697:
5692:
5688:
5684:
5681:
5678:
5675:
5672:
5669:
5639:
5638:
5633:H(X) ⊗ H(X) →
5624:constant sheaf
5601:
5598:
5578:
5577:
5542:
5539:
5538:
5537:
5515:
5460:
5457:
5454:
5448:
5443:
5440:
5432:
5429:
5426:
5423:
5420:
5417:
5394:
5391:
5349:
5346:
5343:
5337:
5334:
5331:
5328:
5325:
5321:
5314:
5311:
5308:
5302:
5299:
5291:
5286:
5283:
5279:
5275:
5272:
5269:
5266:
5263:
5243:
5240:
5237:
5231:
5228:
5225:
5222:
5219:
5216:
5212:
5205:
5202:
5199:
5196:
5191:
5186:
5183:
5179:
5175:
5172:
5169:
5166:
5160:
5157:
5132:
5129:
5119:correspond to
5113:
5112:
5083:
5082:
5056:abelian groups
5045:
5042:
5041:
5040:
5038:partial orders
4975:
4972:
4950:
4949:
4935:
4913:affine schemes
4849:
4846:
4839:
4835:
4832:
4803:
4799:
4796:
4786:
4782:
4778:
4773:
4772:
4768:
4764:
4759:
4758:
4755:
4730:
4726:
4722:
4716:
4712:
4708:
4657:factor modules
4643:
4637:correspond to
4633:in a category
4618:
4614:
4611:
4607:
4603:
4600:
4596:
4592:
4589:
4585:
4582:
4579:
4570:
4567:
4563:
4559:
4556:
4552:
4548:
4545:
4541:
4538:
4534:
4531:
4525:
4522:
4518:
4515:
4506:
4503:
4499:
4496:
4482:Hilbert spaces
4465:
4459:
4433:
4417:
4413:
4409:
4405:
4400:
4399:
4395:
4391:
4386:
4385:
4382:
4362:
4358:
4354:
4352:
4327:
4324:
4319:Main article:
4316:
4313:
4296:
4293:
4288:
4284:
4280:
4277:
4274:
4271:
4268:
4229:
4204:
4201:
4197:
4193:
4190:
4187:
4184:
4181:
4158:
4155:
4154:
4153:
4140:
4137:
4133:
4130:
4125:
4119:
4096:
4091:
4086:
4081:
4077:
4072:
4049:
4046:
4043:
4039:
4034:
4013:
4008:
4002:
3996:
3989:
3986:provided that
3981:
3977:
3970:
3962:
3953:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3888:
3884:
3880:
3877:
3868:defines a map
3837:
3833:
3830:
3826:
3823:
3796:
3793:
3772:
3752:
3749:
3731:
3727:
3724:
3721:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3687:
3682:
3677:
3674:
3670:
3644:
3639:
3634:
3631:
3609:
3604:
3601:
3577:
3573:
3567:
3563:
3557:
3552:
3549:
3546:
3542:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3516:
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3479:
3476:
3473:
3470:
3467:
3442:
3438:
3432:
3427:
3422:
3411:linear algebra
3394:
3391:
3388:
3385:
3382:
3379:
3370:which satisfy
3358:
3354:
3349:
3344:
3339:
3336:
3314:
3310:
3304:
3299:
3294:
3274:
3254:
3232:
3227:
3222:
3219:
3197:
3192:
3189:
3140:
3137:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
2976:
2961:
2947:
2943:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2900:
2896:
2883:
2880:
2869:
2865:
2858:
2854:
2847:
2843:
2839:
2833:
2829:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2769:
2765:
2759:
2755:
2751:
2748:
2743:
2740:
2736:
2732:
2729:
2699:
2695:
2691:
2668:
2664:
2654:
2651:
2628:locally convex
2612:
2609:
2605:
2602:
2577:
2574:
2559:
2544:
2541:
2486:
2482:
2458:
2455:
2454:
2453:
2447:
2446:
2443:
2439:
2435:
2426:
2425:
2422:
2411:
2410:
2407:
2403:
2399:
2389:
2388:
2382:
2359:
2358:
2355:
2345:
2344:
2338:
2328:
2324:
2320:
2316:
2308:set complement
2300:
2299:
2296:temporal logic
2234:
2223:
2214:
2208:
2195:
2194:
2191:
2187:
2179:
2178:
2175:
2167:
2166:
2163:
2148:
2144:
2136:
2132:
2119:
2107:free variables
2092:
2091:holds for all
2088:
2084:
2080:
2076:
2070:
2066:
2062:
2054:
2050:
2046:
2030:
2026:
2022:
2018:
2005:
2002:
1977:
1972:
1958:linear program
1872:
1868:
1808:
1805:
1760:
1742:
1720:
1699:
1696:
1690:
1687:
1681:
1677:
1673:
1667:, that is, an
1647:
1646:
1635:
1620:
1590:
1586:
1575:
1571:
1560:Main article:
1530:
1527:
1526:
1525:
1512:
1501:
1497:
1492:
1486:
1482:
1478:
1474:
1466:
1462:
1445:
1439:
1435:
1428:
1424:
1420:
1414:
1410:
1406:
1395:
1391:
1381:
1377:
1372:. For a fixed
1364:
1361:
1330:
1322:
1321:
1316:
1306:
1290:
1285:
1281:
1275:
1271:
1267:
1250:
1246:
1240:
1221:
1216:
1182:
1179:
1176:
1173:
1170:
1154:
1140:linear algebra
1135:
1132:
1131:
1130:
1116:
1112:
1091:
1069:
1065:
1044:
1035:of the plane,
1024:
1004:
993:
979:
975:
971:
966:
962:
941:
938:
935:
920:
919:
907:
885:
882:
878:
857:
837:
813:
793:
771:
768:
764:
743:
723:
703:
698:
694:
690:
685:
681:
677:
657:
654:
649:
645:
639:
635:
631:
626:
622:
616:
612:
591:
586:
582:
578:
573:
569:
565:
543:
538:
533:
528:
524:
499:
494:
469:
464:
442:
417:
404:
401:
365:closed subsets
353:
352:
347:
340:if and only if
335:
315:
310:
306:
296:
292:
288:
277:
273:
250:
241:
237:
224:
221:
209:Michael Atiyah
204:
201:
143:test functions
26:
9:
6:
4:
3:
2:
7402:
7391:
7388:
7387:
7385:
7374:
7368:
7364:
7359:
7356:
7352:
7348:
7342:
7338:
7334:
7329:
7326:
7322:
7318:
7314:
7310:
7307:
7305:0-07-054235-X
7301:
7297:
7293:
7288:
7285:
7281:
7277:
7273:
7269:
7265:
7261:
7257:
7252:
7249:
7245:
7241:
7235:
7231:
7230:
7224:
7221:
7215:
7211:
7206:
7205:
7198:
7195:
7191:
7187:
7183:
7178:
7173:
7169:
7165:
7161:
7157:
7154:
7148:
7144:
7140:
7139:
7134:
7130:
7126:
7121:
7117:
7112:
7109:
7105:
7101:
7095:
7091:
7087:
7083:
7079:
7076:
7072:
7068:
7062:
7058:
7054:
7049:
7046:
7042:
7038:
7032:
7028:
7024:
7020:
7016:
7009:
7005:
7001:
6997:
6994:
6990:
6986:
6980:
6976:
6972:
6967:
6964:
6960:
6956:
6952:
6948:
6942:
6938:
6934:
6933:
6928:
6924:
6921:
6915:
6911:
6907:
6903:
6899:
6896:
6892:
6888:
6882:
6878:
6874:
6870:
6866:
6862:
6859:
6853:
6849:
6845:
6841:
6837:
6828:on 2021-02-09
6827:
6823:
6819:
6815:
6811:
6807:
6803:
6800:
6799:author's site
6789:on 2011-07-24
6788:
6784:
6780:
6776:
6772:
6767:
6764:
6763:author's site
6758:
6753:
6748:
6743:
6739:
6735:
6731:
6727:
6723:
6722:
6714:
6710:
6709:
6699:
6695:
6691:
6687:
6682:
6677:
6673:
6669:
6665:
6660:
6655:
6654:
6648:
6644:
6640:
6639:
6634:
6630:
6626:
6619:
6614:
6613:
6596:
6592:
6586:
6579:
6573:
6566:
6560:
6553:
6550:Iversen
6547:
6540:
6534:
6528:, p. 56
6527:
6521:
6514:
6511:(Loomis
6508:
6501:
6495:
6488:
6482:
6475:
6469:
6462:
6457:
6450:
6444:
6437:
6434:Weibel (
6431:
6423:
6417:
6413:
6412:
6404:
6398:, §19C)
6397:
6391:
6384:
6383:Mac Lane 1998
6379:
6373:
6367:
6360:
6359:Edwards (1965
6355:
6348:
6342:
6335:
6331:
6330:finite fields
6325:
6318:
6312:
6305:
6300:
6294:
6289:
6283:
6278:
6259:
6251:
6248:
6230:
6227:
6223:
6212:
6206:, p. 34)
6205:
6200:
6181:
6174:
6169:
6162:
6157:
6150:
6145:
6138:
6133:
6126:
6121:
6117:
6106:
6102:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6057:
6054:
6051:
6047:
6043:
6040:
6038:
6035:
6033:
6030:
6028:
6025:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5987:
5984:
5982:
5979:
5977:
5974:
5973:
5966:
5964:
5960:
5959:local duality
5956:
5949:
5945:
5941:
5937:
5933:
5929:
5925:
5921:
5917:
5916:
5915:
5913:
5910:
5908:
5903:
5899:
5880:
5863:
5859:
5855:
5851:
5847:
5843:
5839:
5838:global fields
5835:
5831:
5827:
5822:
5820:
5816:
5812:
5806:
5804:
5800:
5796:
5795:Serre duality
5792:
5788:
5784:
5777:
5773:
5769:
5765:
5755:
5753:
5749:
5746: −
5745:
5741:
5737:
5733:
5729:
5724:
5722:
5718:
5714:
5695:
5690:
5686:
5676:
5673:
5670:
5660:
5659:
5658:
5656:
5652:
5648:
5644:
5636:
5632:
5631:
5630:
5628:
5625:
5621:
5617:
5613:
5610:
5606:
5597:
5595:
5591:
5587:
5583:
5575:
5571:
5567:
5564:
5563:
5562:
5560:
5556:
5552:
5548:
5535:
5531:
5527:
5523:
5519:
5516:
5513:
5509:
5505:
5502:
5501:
5500:
5498:
5494:
5490:
5486:
5482:
5478:
5474:
5455:
5446:
5441:
5438:
5430:
5424:
5421:
5415:
5392:
5389:
5378:
5374:
5370:
5368:
5363:
5347:
5344:
5341:
5335:
5332:
5329:
5326:
5323:
5319:
5309:
5300:
5297:
5281:
5277:
5273:
5267:
5261:
5241:
5238:
5235:
5229:
5226:
5223:
5220:
5217:
5214:
5210:
5200:
5194:
5181:
5177:
5173:
5167:
5158:
5155:
5144:
5140:
5138:
5128:
5127:, see below.
5126:
5122:
5118:
5110:
5106:
5103:
5102:
5101:
5099:
5095:
5092:
5088:
5080:
5076:
5072:
5068:
5067:
5066:
5065:
5061:
5057:
5054:
5050:
5039:
5035:
5031:
5028:
5027:
5026:
5024:
5020:
5017:, connecting
5016:
5015:Stone duality
5012:
5008:
5003:
5001:
4997:
4993:
4989:
4988:Galois theory
4985:
4981:
4971:
4969:
4965:
4961:
4959:
4955:
4947:
4946:
4945:
4943:
4938:
4933:
4929:
4925:
4924:
4918:
4914:
4910:
4906:
4901:
4899:
4895:
4891:
4887:
4883:
4879:
4875:
4871:
4868:
4865:
4861:
4858:
4854:
4831:
4795:
4754:
4753:
4749:
4745:
4741:
4737:
4706:
4701:
4699:
4695:
4691:
4687:
4683:
4679:
4674:
4671:(also called
4670:
4666:
4662:
4658:
4654:
4650:
4640:
4632:
4578:
4530:
4513:
4494:
4490:
4485:
4483:
4479:
4474:
4431:
4427:
4381:
4379:
4375:
4371:
4351:
4342:
4338:
4334:
4322:
4312:
4310:
4294:
4286:
4282:
4278:
4275:
4269:
4266:
4258:
4255:
4252:
4248:
4244:
4218:
4202:
4191:
4188:
4182:
4179:
4168:
4164:
4138:
4131:
4089:
4075:
4044:
4037:
4021:
4017:
4016:Distributions
4014:
3967:
3958:
3954:
3951:
3947:
3943:
3927:
3918:
3915:
3912:
3903:
3894:
3891:
3886:
3882:
3875:
3864:
3863:inner product
3857:
3856:Hilbert space
3853:
3852:
3851:
3848:
3835:
3831:
3828:
3824:
3821:
3813:
3811:
3794:
3791:
3770:
3762:
3758:
3748:
3746:
3729:
3725:
3722:
3719:
3711:
3708:
3702:
3699:
3696:
3690:
3685:
3675:
3672:
3668:
3660:
3642:
3632:
3629:
3607:
3575:
3571:
3565:
3561:
3555:
3550:
3547:
3544:
3540:
3536:
3530:
3527:
3524:
3518:
3505:
3495:
3490:
3480:
3474:
3471:
3468:
3456:
3440:
3430:
3412:
3408:
3392:
3389:
3383:
3377:
3347:
3337:
3334:
3312:
3302:
3272:
3252:
3230:
3220:
3217:
3195:
3176:
3174:
3173:Dual theorems
3170:
3166:
3162:
3158:
3150:
3145:
3136:
3134:
3130:
3126:
3120:
3116:
3110:
3107:
3100:
3095:
3091:
3087:
3083:
3079:
3078:tangent space
3074:
3069:
3065:
3061:
3057:
3052:
3047:
3044:is called an
3042:
3038:In this case
3025:
3019:
3016:
3013:
3010:
3007:
3000:
2999:bilinear form
2995:
2991:precisely if
2989:
2983:
2973:
2967:
2960:
2954:
2940:
2921:
2918:
2915:
2909:
2906:
2903:
2898:
2894:
2879:
2814:
2808:
2805:
2799:
2796:
2793:
2787:
2784:
2778:
2775:
2772:
2767:
2757:
2753:
2746:
2741:
2738:
2734:
2727:
2719:
2707:
2684:
2650:
2648:
2644:
2639:
2637:
2633:
2629:
2610:
2607:
2600:
2575:
2572:
2542:
2539:
2528:
2524:
2520:
2515:
2513:
2509:
2505:
2501:
2497:
2493:
2478:
2474:
2472:
2468:
2464:
2373:: an element
2309:
2305:
2304:
2303:
2297:
2245:
2219:
2215:
2211:
2207:
2203:
2156:
2120:
2116:
2112:
2108:
2104:
2103:
2098:
2097:
2096:
2042:
2038:
2035:, where ¬ is
2001:
1999:
1995:
1975:
1959:
1955:
1950:
1948:
1940:
1932:
1924:
1919:
1915:
1911:
1903:
1899:
1894:
1890:
1888:
1884:
1880:
1879:face lattices
1856:
1852:
1848:
1844:
1840:
1835:
1831:
1827:
1823:
1813:
1804:
1802:
1798:
1794:
1790:
1787:In topology,
1785:
1783:
1779:
1775:
1771:
1767:
1757:
1749:
1739:
1730:
1718:
1710:
1706:
1670:
1663:
1656:
1652:
1644:
1640:
1639:descendant-of
1636:
1633:
1629:
1625:
1621:
1618:
1602:
1601:
1600:
1598:
1583:
1569:
1563:
1555:
1539:
1538:Hasse diagram
1535:
1523:
1493:
1460:An extension
1459:
1458:
1457:
1454:
1388:
1375:
1371:
1370:Galois theory
1363:Galois theory
1360:
1358:
1354:
1350:
1349:Hilbert space
1338:
1291:
1269:A linear map
1268:
1265:
1261:
1237:
1236:
1235:
1219:
1200:
1180:
1174:
1171:
1168:
1161:
1152:
1145:
1141:
1114:
1110:
1089:
1067:
1063:
1042:
1022:
1002:
994:
977:
973:
969:
964:
960:
939:
936:
933:
925:
924:
923:
905:
883:
880:
876:
855:
835:
827:
826:
825:
811:
791:
769:
766:
762:
741:
721:
696:
692:
688:
683:
679:
655:
652:
647:
643:
637:
633:
629:
624:
620:
614:
610:
584:
580:
576:
571:
567:
541:
531:
526:
522:
497:
467:
440:
432:
428:
425:A duality in
409:
400:
390:
386:
366:
362:
358:
341:
316:
303:
289:
286:
285:
269:
268:
267:
248:
230:
220:
216:
212:
210:
200:
197:
193:
189:
183:
179:
175:
170:
166:
162:
157:
155:
151:
150:
145:
144:
140:
139:distributions
134:
130:
126:
121:
118:
113:
111:
110:
106:
100:
96:
84:
80:
60:
56:
52:
48:
44:
40:
33:
19:
7362:
7332:
7316:
7291:
7259:
7255:
7228:
7203:
7167:
7163:
7160:Mazur, Barry
7137:
7115:
7085:
7052:
7014:
7004:Street, Ross
7000:Joyal, André
6970:
6931:
6905:
6872:
6843:
6830:, retrieved
6826:the original
6813:
6791:, retrieved
6787:the original
6782:
6778:
6737:
6733:
6671:
6667:
6652:
6636:
6589:Mazur (
6585:
6576:Milne (
6572:
6559:
6546:
6533:
6520:
6507:
6494:
6481:
6468:
6456:
6443:
6430:
6410:
6403:
6390:
6378:
6370:Fulton
6366:
6354:
6341:
6324:
6311:
6299:
6288:
6277:
6211:
6199:
6180:
6168:
6156:
6144:
6132:
6120:
6049:
6044:, a certain
6042:Dual numbers
6032:Dual lattice
5952:
5947:
5943:
5939:
5935:
5931:
5927:
5923:
5919:
5911:
5906:
5901:
5861:
5857:
5853:
5850:Galois group
5823:
5813:and certain
5807:
5775:
5761:
5747:
5743:
5735:
5732:cell complex
5725:
5720:
5716:
5712:
5710:
5646:
5642:
5640:
5634:
5626:
5615:
5611:
5603:
5593:
5585:
5579:
5573:
5569:
5565:
5558:
5550:
5544:
5492:
5488:
5484:
5480:
5376:
5372:
5366:
5361:
5141:
5134:
5114:
5108:
5104:
5093:
5091:circle group
5086:
5084:
5078:
5074:
5070:
5059:
5047:
5021:and spatial
5019:sober spaces
5006:
5004:
4999:
4995:
4991:
4977:
4962:
4951:
4936:
4927:
4922:
4916:
4902:
4897:
4889:
4885:
4881:
4877:
4873:
4869:
4859:
4851:
4809:
4793:
4751:
4747:
4743:
4739:
4702:
4694:cofibrations
4653:monomorphism
4649:epimorphisms
4625:for any set
4624:
4576:
4486:
4477:
4475:
4423:
4380:gives a map
4377:
4373:
4369:
4367:
4339:between two
4329:
4259:is given by
4256:
4172:is given by
4163:dual lattice
4160:
3849:
3814:
3808:is called a
3754:
3457:
3177:
3154:
3118:
3114:
3105:
3098:
3072:
3050:
3040:
2993:
2987:
2981:
2978:
2971:
2965:
2885:
2863:) the value
2717:
2705:
2683:internal Hom
2656:
2653:Dual objects
2640:
2516:
2479:
2475:
2470:
2466:
2462:
2460:
2301:
2209:
2205:
2201:
2114:
2110:
2100:
2007:
1998:dual problem
1951:
1910:planar graph
1907:
1898:planar graph
1818:
1801:dual matroid
1786:
1738:dual concept
1737:
1731:
1650:
1648:
1642:
1638:
1627:
1623:
1565:
1494:Associating
1455:
1387:Galois group
1366:
1352:
1323:
1144:vector space
1137:
921:
898:. (For some
424:
354:
301:
282:
226:
218:
214:
206:
195:
191:
187:
181:
177:
173:
158:
147:
136:
132:
122:
116:
114:
102:
98:
83:fixed points
78:
42:
36:
7296:McGraw-Hill
7166:, Série 4,
7082:Lang, Serge
6537:Milne
6385:, Ch. II.1.
6204:Rudin (1976
6175:, p. 1
6173:Atiyah 2007
6161:Gowers 2008
6149:Gowers 2008
6127:, p. 1
6125:Atiyah 2007
6056:Dual system
5986:Convex body
5826:arithmetics
5547:dual spaces
5473:convolution
4857:C*-algebras
4669:direct sums
4307:continuous
3944:due to the
3940:which is a
2939:linear maps
2521:(including
2508:dual module
2498:" from the
2467:double dual
2314:. That is,
2274:is true in
2258:is true in
2218:modal logic
2102:satisfiable
2041:quantifiers
1887:involutions
1830:convex hull
1793:closed sets
1705:composition
1671:involution
1643:ancestor-of
1628:multiple-of
1264:isomorphism
1254:. For some
1201:over which
754:. Instead,
602:satisfying
371:: a subset
39:mathematics
7372:0030505356
6832:2009-03-11
6793:2009-05-30
6605:References
6394:(Lam
5997:Dual basis
5752:cohomology
5522:velocities
5115:Moreover,
4690:fibrations
4678:projective
4673:coproducts
4661:submodules
4341:categories
3850:Examples:
3659:orthogonal
3090:Hodge star
2529:, denoted
2262:", while "
2254:such that
2083:for which
1914:dual graph
1902:dual graph
1774:upper sets
1770:lower sets
1754:will be a
304:direction
284:involution
247:complement
59:involution
55:one-to-one
7276:0021-8693
7186:0012-9593
6742:CiteSeerX
6690:0002-9904
6643:EMS Press
6633:"Duality"
6631:(2001) ,
6361:, 8.4.7).
6231:∗
6228:∗
6101:T-duality
6096:S-duality
6037:Dual norm
6007:Dual code
5881:^
5805:instead.
5696:ω
5691:γ
5687:∫
5683:↦
5677:ω
5671:γ
5551:dualities
5508:operators
5447:^
5442:^
5422:−
5393:^
5369:-function
5345:ξ
5336:ξ
5327:π
5310:ξ
5301:^
5290:∞
5285:∞
5282:−
5278:∫
5230:ξ
5221:π
5215:−
5190:∞
5185:∞
5182:−
5178:∫
5168:ξ
5159:^
5073:) = Hom (
5032:relating
4610:) × Hom (
4599:) = Hom (
4562:) × Hom (
4551:) = Hom (
4478:self-dual
4270:
4183:
4124:∞
3942:bijection
3922:⟩
3910:⟨
3907:↦
3898:↦
3887:∗
3879:→
3825:≅
3723:∈
3706:⟩
3694:⟨
3676:∈
3633:⊂
3541:∑
3534:⟩
3522:⟨
3511:→
3496:×
3478:⟩
3475:⋅
3469:⋅
3466:⟨
3441:∗
3353:→
3313:∗
3221:⊂
3096:. For an
3023:→
3017:×
3008:φ
2953:dimension
2910:
2899:∗
2788:
2779:
2768:∗
2758:∗
2742:∗
2739:∗
2731:→
2604:→
2494:, and a "
2310:operator
1994:halfspace
1789:open sets
1717:power set
1554:upper set
1542:{1,2,3,4}
1178:→
1169:φ
1115:∗
1068:∗
978:∗
970:⊆
965:∗
937:⊆
884:∗
881:∗
770:∗
767:∗
653:≥
532:⊆
527:∗
431:dual cone
403:Dual cone
185:its dual
103:standard
99:self-dual
18:Self-dual
7384:Category
7135:(1998),
7084:(2002),
7006:(1991),
6963:13348052
6929:(1977),
6904:(1966),
6871:(1994),
6842:(1993),
6773:(2007),
6332:or even
5969:See also
5785:, using
5770:, using
5740:homology
5728:manifold
5137:analysis
4907:between
4894:spectrum
4705:functors
4639:colimits
4139:′
4076:′
4038:′
4006:. Hence
3832:″
3795:″
3155:In some
3129:magnetic
3082:manifold
2842: :
2611:″
2576:″
2543:′
1937:and the
1834:vertices
1676: :
1632:integers
1566:Given a
1193:, where
427:geometry
385:interior
357:topology
302:opposite
169:pullback
125:pairings
47:theorems
7355:1269324
7325:0179666
7284:0280571
7248:2261462
7194:0344254
7108:1878556
7086:Algebra
7075:1653294
7045:1173027
6993:0842190
6955:0463157
6895:1288523
6822:1361887
6797:. Also
6761:. Also
6698:1848254
5930:, Hom (
5926:) × H (
5909:-module
5766:over a
5754:group.
5622:of the
5588:to the
5557:of two
5530:momenta
5089:to the
5023:locales
4954:schemes
4892:as the
4864:compact
4798:colim:
4771:) ≅ Hom
4736:adjoint
4398:) → Hom
4167:lattice
2716:to the
2694:= Hom (
2624:
2593:
2589:
2564:
2556:
2531:
1904:in red.
1797:matroid
1782:filters
1624:divides
1469:′
1404:(i.e.,
1197:is the
389:closure
229:subsets
165:functor
159:From a
117:duality
105:duality
43:duality
7369:
7353:
7343:
7323:
7302:
7282:
7274:
7246:
7236:
7216:
7192:
7184:
7149:
7106:
7096:
7073:
7063:
7043:
7033:
6991:
6981:
6961:
6953:
6943:
6916:
6893:
6883:
6854:
6820:
6744:
6696:
6688:
6418:
6250:convex
6247:closed
6050:double
5896:, the
5830:finite
5641:where
5364:is an
5316:
5207:
5107:≅ χ(χ(
5062:, the
4631:limits
4441:, and
4245:. The
3992:< ∞
3974:where
3966:-space
3959:of an
3866:⟨⋅, ⋅⟩
3109:-forms
3062:, any
2718:bidual
2471:primal
2463:bidual
2457:Bidual
1845:, any
1778:ideals
1776:, and
1707:is an
1653:is an
1584:poset
1423:= Gal(
421:(red).
411:A set
245:, the
79:primal
7011:(PDF)
6621:(PDF)
6112:Notes
5942:)) →
5834:local
5774:with
4921:Spec
4842:: lim
4581:Hom (
4533:Hom (
4487:Many
4165:of a
3080:of a
2663:Hom (
2238:that
2171:) = ¬
2115:every
2111:valid
2025:) = ¬
1660:of a
1617:above
1568:poset
1347:is a
1337:basis
1199:field
7367:ISBN
7341:ISBN
7300:ISBN
7272:ISSN
7234:ISBN
7214:ISBN
7182:ISSN
7147:ISBN
7094:ISBN
7061:ISBN
7031:ISBN
6979:ISBN
6959:OCLC
6941:ISBN
6914:ISBN
6881:ISBN
6852:ISBN
6686:ISSN
6595:2006
6591:1973
6578:2006
6565:1966
6552:1986
6539:1980
6526:1994
6513:1953
6500:2002
6487:1991
6474:1966
6449:1995
6436:1994
6416:ISBN
6396:1999
6372:1993
6345:See
6317:1965
5836:and
5750:)th
5653:and
5590:dual
5580:For
5528:and
5483:(or
5408:and
5036:and
4911:and
4862:and
4746:and
4734:are
4720:and
4703:Two
4692:and
4680:and
4667:vs.
4651:vs.
4577:and
4510:and
4469:and
4457:and
4449:and
4372:and
4346:and
4161:The
3988:1 ≤
3980:+ 1/
3955:The
3861:its
3657:its
3147:The
3131:and
3056:real
2969:and
2853:Hom(
2706:dual
2517:For
2369:and
2290:and
2282:and
2141:and
2125:and
2059:and
2012:and
1875:− 1)
1791:and
1780:and
1772:and
1641:and
1637:the
1626:and
1622:the
1607:and
1593:, ≥)
1582:dual
1578:, ≤)
1510:and
1496:Gal(
1473:Gal(
1390:Gal(
1355:the
1328:and
1296:and
1149:its
1015:and
363:and
361:open
321:and
276:) =
41:, a
7264:doi
7172:doi
7120:hdl
7023:doi
6752:doi
6738:254
6676:doi
5965:).
5900:of
5797:or
5738:th
5592:of
5532:in
5524:in
5375:or
5371:on
5360:If
5135:In
5111:)).
4896:of
4880:to
4834:Δ:
4822:of
4806:: Δ
4767:),
4763:(F(
4757:Hom
4750:in
4742:in
4684:in
4437:of
4412:),
4384:Hom
4376:of
4267:Hom
4249:of
4180:Hom
3984:= 1
3968:is
3409:of
3111:to
3058:or
2955:of
2907:Hom
2785:Hom
2776:Hom
2708:of
2645:(a
2465:or
2351:= (
2323:= (
2246:, "
2216:In
2162:(¬
2135:∧ ¬
2113:if
1941:of
1933:of
1758:of
1750:of
1723:= 2
1711:of
1589:= (
1574:= (
1516:to
1506:to
1353:via
1339:of
1314:to
1304:to
714:in
375:of
325:of
107:in
97:is
89:is
73:is
65:is
49:or
37:In
7386::
7351:MR
7349:,
7339:,
7335:,
7321:MR
7298:,
7280:MR
7278:,
7270:,
7260:19
7258:,
7244:MR
7242:,
7212:,
7208:,
7190:MR
7188:,
7180:,
7145:,
7104:MR
7102:,
7092:,
7071:MR
7069:,
7059:,
7041:MR
7039:,
7029:,
7013:,
7002:;
6989:MR
6987:,
6977:,
6957:,
6951:MR
6949:,
6939:,
6935:,
6891:MR
6889:,
6879:,
6867:;
6850:,
6846:,
6818:MR
6812:,
6783:14
6781:,
6777:,
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