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71-547: [REDACTED] Look up dual in Wiktionary, the free dictionary. Dual or Duals may refer to: Paired/two things [ edit ] Dual (mathematics) , a notion of paired concepts that mirror one another Dual (category theory) , a formalization of mathematical duality see more cases in Category:Duality theories Dual (grammatical number) ,

142-422: A dual graph , 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 graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by

213-440: A dual polyhedron or dual polytope, with an i -dimensional feature of an n -dimensional polytope corresponding to an ( n − i − 1) -dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals . Duality of polytopes and order-theoretic duality are both involutions :

284-597: A real vector space V is a pairing (set M = N = V , R = R in the above definitions). The determinant map (2 × 2 matrices over k ) → k can be seen as a pairing k 2 × k 2 → k {\displaystyle k^{2}\times k^{2}\to k} . The Hopf map S 3 → S 2 {\displaystyle S^{3}\to S^{2}} written as h : S 2 × S 2 → S 2 {\displaystyle h:S^{2}\times S^{2}\to S^{2}}

355-409: A "principle". 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. A simple duality arises from considering subsets of a fixed set S . To any subset A ⊆ S , the complement A consists of all those elements in S that are not contained in A . It is again a subset of S . Taking

426-490: A 2013 EP by Sampha Duals , an album by U2 The Duals , American duo See also [ edit ] [REDACTED] Search for "dual" on Misplaced Pages. Duality (disambiguation) Duel (disambiguation) , a homonym All pages with titles containing Dual or Duals All pages with titles beginning with Dual Double (disambiguation) Duo (disambiguation) Pair (disambiguation) Twin (disambiguation) Topics referred to by

497-527: A close relation between objects of seemingly different nature. One example of such a more general duality is from Galois theory . For a fixed Galois extension K / F , one may associate the Galois group Gal( K / E ) to any intermediate field E (i.e., F ⊆ E ⊆ K ). This group is a subgroup of the Galois group G = Gal( K / F ) . Conversely, to any such subgroup H ⊆ G there

568-815: A consequence of the dimension formula of linear algebra , this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to ( R 3 ) ∗ {\displaystyle (\mathbb {R} ^{3})^{*}} . The (positive definite) bilinear form ⟨ ⋅ , ⋅ ⟩ : R 3 × R 3 → R , ⟨ x , y ⟩ = ∑ i = 1 3 x i y i {\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}} yields an identification of this projective plane with

639-554: 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 computational geometry : the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S . As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in

710-470: A grammatical category used in some languages Dual county , a Gaelic games county which in both Gaelic football and hurling Dual diagnosis , a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem Dual fertilization, simultaneous application of a P-type and N-type fertilizer Dual impedance , electrical circuits that are the dual of each other Dual SIM cellphone supporting use of two SIMs Aerochute International Dual

781-434: A map That functor may or may not be an equivalence of categories . There are various situations, where such a functor is an equivalence between the opposite category C of C , and D . 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 C and D

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852-572: A multiplicative group , all of prime order p {\displaystyle \textstyle p} . Let P ∈ G 1 , Q ∈ G 2 {\displaystyle \textstyle P\in G_{1},Q\in G_{2}} be generators of G 1 {\displaystyle \textstyle G_{1}} and G 2 {\displaystyle \textstyle G_{2}} respectively. A pairing

923-454: A natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagram which has c at all places. Dually, Gelfand duality is a duality between commutative C*-algebras A and compact Hausdorff spaces X

994-406: A pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor , at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f : V → W its dual f : W → V . In the words of Michael Atiyah , Duality in mathematics is not a theorem, but

1065-419: A pairing is an R -linear map where M ⊗ R N {\displaystyle M\otimes _{R}N} denotes the tensor product of M and N . A pairing can also be considered as an R -linear map Φ : M → Hom R ⁡ ( N , L ) {\displaystyle \Phi :M\to \operatorname {Hom} _{R}(N,L)} , which matches

1136-437: A point, in an incidence-preserving way. For such planes there arises a general principle of duality in projective planes : 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

1207-533: A special one-row and one-column database table Dual-Ghia , US-brand of luxury-car of the late 1950s Media [ edit ] Dual (2008 film) , a 2008 western drama film Dual (2022 film) , a 2022 science fiction thriller film "Dual" ( Heroes ) , an episode of Heroes Dual! Parallel Trouble Adventure , an anime series Dual (album) , an album of traditional Scottish and Irish music recorded by Éamonn Doorley, Muireann Nic Amhlaoibh, Julie Fowlis and Ross Martin, released 2008 Dual (EP) ,

1278-676: A structure similar to that of X . This is sometimes called internal Hom . In general, this yields a true duality only for specific choices of D , in which case X = Hom ( X , D ) is referred to as the dual of X . There is always a map from X to the bidual , that is to say, the dual of the dual, X → X ∗ ∗ := ( X ∗ ) ∗ = Hom ⁡ ( Hom ⁡ ( X , D ) , D ) . {\displaystyle X\to X^{**}:=(X^{*})^{*}=\operatorname {Hom} (\operatorname {Hom} (X,D),D).} It assigns to some x ∈ X

1349-503: A system of linear constraints (specifying that the point lie in a halfspace ; 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 dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa. In logic, functions or relations A and B are considered dual if A (¬ x ) = ¬ B ( x ) , where ¬

1420-508: A two-seat Australian powered parachute design Acronyms and other uses [ edit ] Dual (brand) , a manufacturer of Hifi equipment DUAL (cognitive architecture) , an artificial intelligence design model DUAL algorithm , or diffusing update algorithm, used to update Internet protocol routing tables Dual language, alternative spelling of the Australian Aboriginal Dhuwal language DUAL table ,

1491-512: A unique point, the intersection point of these two lines". For further examples, see Dual theorems . 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 R P 2 {\displaystyle \mathbb {RP} ^{2}} correspond to one-dimensional subvector spaces V ⊂ R 3 {\displaystyle V\subset \mathbb {R} ^{3}} while

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1562-525: Is logical negation . The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃ x .¬ P ( x ) and ¬∀ x . P ( x ) are equivalent for all predicates P in classical logic: if there exists an x for which P fails to hold, then it is false that P holds for all x (but the converse does not hold constructively). From this fundamental logical duality follow several others: Other analogous dualities follow from these: The dual of

1633-460: Is a map: e : G 1 × G 2 → G T {\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}} for which the following holds: Note that it is also common in cryptographic literature for all groups to be written in multiplicative notation. In cases when G 1 = G 2 = G {\displaystyle \textstyle G_{1}=G_{2}=G} ,

1704-638: Is an order automorphism of S ; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set S = 2 are induced by permutations of R . A concept defined for a partial order P will correspond to a dual concept on the dual poset P . For instance, a minimal element of P will be a maximal element of P : minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds , lower sets and upper sets , and ideals and filters . In topology, open sets and closed sets are dual concepts:

1775-476: Is an affine spectrum, Spec A . Conversely, given an affine scheme S , one gets back a ring by taking global sections of the structure sheaf O S . In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence Pairing In mathematics , a pairing is an R - bilinear map from the Cartesian product of two R - modules , where

1846-400: Is an example of a pairing. For instance, Hardie et al. present an explicit construction of the map using poset models. In cryptography , often the following specialized definition is used: Let G 1 , G 2 {\displaystyle \textstyle G_{1},G_{2}} be additive groups and G T {\displaystyle \textstyle G_{T}}

1917-442: Is an important concept in elliptic curve cryptography ; e.g., it may be used to attack certain elliptic curves (see MOV attack ). It and other pairings have been used to develop identity-based encryption schemes. Scalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example, in representation theory , one has a scalar product on the characters of complex representations of

1988-788: Is because for a generator g ∈ G {\displaystyle g\in G} , there exist integers p {\displaystyle p} , q {\displaystyle q} such that P = g p {\displaystyle P=g^{p}} and Q = g q {\displaystyle Q=g^{q}} . Therefore e ( P , Q ) = e ( g p , g q ) = e ( g , g ) p q = e ( g q , g p ) = e ( Q , P ) {\displaystyle e(P,Q)=e(g^{p},g^{q})=e(g,g)^{pq}=e(g^{q},g^{p})=e(Q,P)} . The Weil pairing

2059-1163: Is called non-degenerate on the left if e ( m , n ) = 0 {\displaystyle e(m,n)=0} for all n {\displaystyle n} implies m = 0 {\displaystyle m=0} . A pairing is called alternating if N = M {\displaystyle N=M} and e ( m , m ) = 0 {\displaystyle e(m,m)=0} for all m . In particular, this implies e ( m + n , m + n ) = 0 {\displaystyle e(m+n,m+n)=0} , while bilinearity shows e ( m + n , m + n ) = e ( m , m ) + e ( m , n ) + e ( n , m ) + e ( n , n ) = e ( m , n ) + e ( n , m ) {\displaystyle e(m+n,m+n)=e(m,m)+e(m,n)+e(n,m)+e(n,n)=e(m,n)+e(n,m)} . Thus, for an alternating pairing, e ( m , n ) = − e ( n , m ) {\displaystyle e(m,n)=-e(n,m)} . Any scalar product on

2130-405: Is called a reflexive space : X ≅ X ″ . {\displaystyle X\cong X''.} Examples: The dual lattice of a lattice L is given by Hom ⁡ ( L , Z ) , {\displaystyle \operatorname {Hom} (L,\mathbb {Z} ),} the set of linear functions on the real vector space containing the lattice that map

2201-414: Is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U . A duality in geometry is provided by the dual cone construction. Given a set C {\displaystyle C} of points in

Dual - Misplaced Pages Continue

2272-412: Is defined. The three properties of the dual cone carry over to this type of duality by replacing subsets of R 2 {\displaystyle \mathbb {R} ^{2}} by vector space and inclusions of such subsets by linear maps. That is: A particular feature of this duality is that V and V are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this

2343-530: Is different from Wikidata All article disambiguation pages All disambiguation pages Dual (mathematics) In mathematical contexts, duality 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". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of

2414-407: Is formally the same as an equivalence between C and D ( C and D ). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept. A category that is equivalent to its dual is called self-dual . An example of self-dual category is the category of Hilbert spaces . Many category-theoretic notions come in pairs in

2485-535: Is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a basis of V . This is also true in the case if V is a Hilbert space , via the Riesz representation theorem . 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

2556-495: Is isomorphic to V precisely if V is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate bilinear form φ : V × V → K {\displaystyle \varphi :V\times V\to K} In this case V is called an inner product space . For example, if K is the field of real or complex numbers , any positive definite bilinear form gives rise to such an isomorphism. In Riemannian geometry , V

2627-466: Is taken to be the tangent space of a manifold and such positive bilinear forms are called Riemannian metrics . 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 Hodge star which provides a correspondence between the elements of the exterior algebra . For an n -dimensional vector space,

2698-420: Is the fixed field K consisting of elements fixed by the elements in H . Compared to the above, this duality has the following features: Given a poset P = ( X , ≤) (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 dual poset P = ( X , ≥) comprises the same ground set but

2769-457: Is the same: it assigns to X the space of continuous functions (which vanish at infinity) from X to C , the complex numbers. Conversely, the space X can be reconstructed from A as the spectrum of A . Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way. In a similar vein there is a duality in algebraic geometry between commutative rings and affine schemes : to every commutative ring A there

2840-410: Is weaker than the one above, in that The other two properties carry over without change: A very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V . Its elements are the linear functionals φ : V → K {\displaystyle \varphi :V\to K} , where K is the field over which V

2911-689: The R P 2 {\displaystyle \mathbb {RP} ^{2}} . Concretely, the duality assigns to V ⊂ R 3 {\displaystyle V\subset \mathbb {R} ^{3}} its orthogonal { w ∈ R 3 , ⟨ v , w ⟩ = 0  for all  v ∈ V } {\displaystyle \left\{w\in \mathbb {R} ^{3},\langle v,w\rangle =0{\text{ for all }}v\in V\right\}} . The explicit formulas in duality in projective geometry arise by means of this identification. In

Dual - Misplaced Pages Continue

2982-449: The converse relation . Familiar examples of dual partial orders include A duality transform is an involutive antiautomorphism f of a partially ordered set S , that is, an order-reversing involution f  : S → S . In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if f 1 , f 2 are two duality transforms then their composition

3053-470: The strong dual space topology) as a reflexive space . In other cases, showing a relation between the primal and bidual is a significant result, as in Pontryagin duality (a locally compact abelian group is naturally isomorphic to its bidual). A group of dualities can be described by endowing, for any mathematical object X , the set of morphisms Hom ( X , D ) into some fixed object D , with

3124-421: 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 § Dual objects , and a " natural transformation " from the identity functor to the double dual functor. For vector spaces (considered algebraically), this is always an injection; see Dual space § Injection into

3195-468: The Hodge star operator maps k -forms to ( n − k ) -forms. This can be used to formulate Maxwell's equations . In this guise, the duality inherent in the inner product space exchanges the role of magnetic and electric fields . In some projective planes , it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to

3266-456: The algebraic dual V , with different possible topologies on the dual, each of which defines a different bidual space ⁠ V ″ {\displaystyle V''} ⁠ . In these cases the canonical evaluation map ⁠ V → V ″ {\displaystyle V\to V''} ⁠ is not in general an isomorphism. If it is, this is known (for certain locally convex vector spaces with

3337-401: 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. An important case is for vector spaces, where there is a map from the primal space to the double dual, V → V , known as

3408-480: The circle (with multiplication of complex numbers as group operation). 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 category theory , this amounts to a contravariant functor between two categories C and D : which for any two objects X and Y of C gives

3479-419: The complement has the following properties: This duality appears in topology as a duality between open and closed subsets of some fixed topological space X : a subset U of X is closed if and only if its complement in X 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

3550-411: The complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid . 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

3621-479: 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 C {\displaystyle C} . Instead, C ∗ ∗ {\displaystyle C^{**}} is the smallest cone containing C {\displaystyle C} which may be bigger than C {\displaystyle C} . Therefore this duality

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3692-486: The double-dual . This can be generalized algebraically to a dual module . There is still a canonical evaluation map, but it is not always injective; if it is, this is known as a torsionless module ; if it is an isomophism, the module is called reflexive. For topological vector spaces (including normed vector spaces ), there is a separate notion of a topological dual , denoted ⁠ V ′ {\displaystyle V'} ⁠ to distinguish from

3763-420: 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. From any three-dimensional polyhedron, one can form a planar graph , the graph of its vertices and edges. The dual polyhedron has

3834-416: The dual, called the bidual or double dual , depending on context, is often identical to the original (also called primal ), 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. In other cases,

3905-469: 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 projective and injective modules in homological algebra , fibrations and cofibrations in topology and more generally model categories . Two functors F : C → D and G : D → C are adjoint if for all objects c in C and d in D in

3976-465: 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 other, so do the corresponding two parts of the dual polyhedron . More generally, using the concept of polar reciprocation , any convex polyhedron , or more generally any convex polytope , corresponds to

4047-654: The first definition by setting Φ ( m ) ( n ) := e ( m , n ) {\displaystyle \Phi (m)(n):=e(m,n)} . A pairing is called perfect if the above map Φ {\displaystyle \Phi } is an isomorphism of R -modules. A pairing is called non-degenerate on the right if for the above map we have that e ( m , n ) = 0 {\displaystyle e(m,n)=0} for all m {\displaystyle m} implies n = 0 {\displaystyle n=0} ; similarly, e {\displaystyle e}

4118-405: The introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps , forms a vector space in its own right. The map V → V mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the dimension of V is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis. A vector space V

4189-641: The lines in the projective plane correspond to subvector spaces W {\displaystyle W} of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional V {\displaystyle V} the subspace of ( R 3 ) ∗ {\displaystyle (\mathbb {R} ^{3})^{*}} consisting of those linear maps f : R 3 → R {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} } which satisfy f ( V ) = 0 {\displaystyle f(V)=0} . As

4260-506: The map that associates to any map f  : X → D (i.e., an element in Hom( X , D ) ) the value f ( x ) . Depending on the concrete duality considered and also depending on the object X , this map may or may not be an isomorphism. The construction of the dual vector space V ∗ = Hom ⁡ ( V , K ) {\displaystyle V^{*}=\operatorname {Hom} (V,K)} mentioned in

4331-482: The objects. A classical example of this is the duality of the Platonic solids , 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 dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with

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4402-417: The pairing is called symmetric. As G {\displaystyle \textstyle G} is cyclic , the map e {\displaystyle e} will be commutative ; that is, for any P , Q ∈ G {\displaystyle P,Q\in G} , we have e ( P , Q ) = e ( Q , P ) {\displaystyle e(P,Q)=e(Q,P)} . This

4473-855: The plane R 2 {\displaystyle \mathbb {R} ^{2}} (or more generally points in R n {\displaystyle \mathbb {R} ^{n}} ), the dual cone is defined as the set C ∗ ⊆ R 2 {\displaystyle C^{*}\subseteq \mathbb {R} ^{2}} consisting of those points ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} satisfying x 1 c 1 + x 2 c 2 ≥ 0 {\displaystyle x_{1}c_{1}+x_{2}c_{2}\geq 0} for all points ( c 1 , c 2 ) {\displaystyle (c_{1},c_{2})} in C {\displaystyle C} , as illustrated in

4544-419: The points of the lattice to the integers Z {\displaystyle \mathbb {Z} } . This is used in the construction of toric varieties . The Pontryagin dual of locally compact topological groups G is given by Hom ⁡ ( G , S 1 ) , {\displaystyle \operatorname {Hom} (G,S^{1}),} continuous group homomorphisms with values in

4615-411: 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. Matroid duality is an algebraic extension of planar graph duality, in

4686-419: The realm of topological vector spaces , a similar construction exists, replacing the dual by the topological dual 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 X {\displaystyle X} that is canonically isomorphic to its bidual X ″ {\displaystyle X''}

4757-405: The same term [REDACTED] This disambiguation page lists articles associated with the title Dual . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Dual&oldid=1075817928 " Category : Disambiguation pages Hidden categories: Short description

4828-410: The second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number , viewed as

4899-485: The sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph. A kind of geometric duality also occurs in optimization theory , but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} ),

4970-641: The sense that they correspond to each other while considering the opposite category. For example, Cartesian products Y 1 × Y 2 and disjoint unions Y 1 ⊔ Y 2 of sets are dual to each other in the sense that and for any set X . This is a particular case of a more general duality phenomenon, under which limits in a category C correspond to colimits in the opposite category C ; further concrete examples of this are epimorphisms vs. monomorphism , in particular factor modules (or groups etc.) vs. submodules , direct products vs. direct sums (also called coproducts to emphasize

5041-806: The underlying ring R is commutative . Let R be a commutative ring with unit , and let M , N and L be R -modules . A pairing is any R -bilinear map e : M × N → L {\displaystyle e:M\times N\to L} . That is, it satisfies for any r ∈ R {\displaystyle r\in R} and any m , m 1 , m 2 ∈ M {\displaystyle m,m_{1},m_{2}\in M} and any n , n 1 , n 2 ∈ N {\displaystyle n,n_{1},n_{2}\in N} . Equivalently,

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