Inmathematics, aduality translates concepts,theorems ormathematical structures into other concepts, theorems or structures in aone-to-one fashion, often (but not always) by means of aninvolution operation: if the dual ofA isB, then the dual ofB isA. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also calledprimal). Such involutions sometimes havefixed points, so that the dual ofA isA itself. For example,Desargues' theorem isself-dual in this sense under thestandardduality inprojective geometry.

In mathematical contexts,duality has numerous meanings.^[1] It has been described as "a very pervasive and important concept in (modern) mathematics"^[2] and "an important general theme that has manifestations in almost every area of mathematics".^[3]

Many mathematical dualities between objects of two types correspond topairings,bilinear functions from an object of one type and another object of 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, theduality betweendistributions and the associatedtest functions corresponds to the pairing in which one integrates a distribution against a test function, andPoincaré duality corresponds similarly tointersection number, viewed as a pairing between submanifolds of a given manifold.^[4]

From acategory theory viewpoint, duality can also be seen as afunctor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and thepullback construction assigns to each arrowf:V →W its dualf^∗:W^∗ →V^∗.

In the words ofMichael Atiyah,

Duality in mathematics is not a theorem, but a "principle".^[5]

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 consideringsubsets of a fixed setS. To any subsetA ⊆S, thecomplementA^∁[6] consists of all those elements inS that are not contained inA. It is again a subset ofS. Taking the complement has the following properties:

• Applying it twice gives back the original set, i.e.,(A^∁)^∁ =A. This is referred to by saying that the operation of taking the complement is aninvolution.
• An inclusion of setsA ⊆B is turned into an inclusion in theopposite directionB^∁ ⊆A^∁.
• Given two subsetsA andB ofS,A is contained inB^∁if and only ifB is contained inA^∁.

This duality appears intopology as a duality betweenopen andclosed subsets of some fixed topological spaceX: a subsetU ofX is closed if and only if its complement inX 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.^[7] Theinterior of a set is the largest open set contained in it, and theclosure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any setU is equal to the closure of the complement ofU.

A duality ingeometry is provided by thedual cone construction. Given a set${\displaystyle C}$ of points in the plane${\displaystyle {\mathbb{R}}^2}$ (or more generally points in${\displaystyle {\mathbb{R}}^n}$), the dual cone is defined as the set${\displaystyle C^{\ast }\subseteq {\mathbb{R}}^2}$ consisting of those points${\displaystyle (x_1,x_2)}$ satisfying$${\displaystyle x_1{c}_1+x_2{c}_2\ge 0}$$for all points${\displaystyle (c_1,c_2)}$ in${\displaystyle C}$, 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${\displaystyle C}$. Instead,${\displaystyle C^{\ast \ast }}$ is the smallest cone^[8] containing${\displaystyle C}$ which may be bigger than${\displaystyle C}$. Therefore this duality is weaker than the one above, in that

• Applying the operation twice gives back a possibly bigger set: for all${\displaystyle C}$,${\displaystyle C}$ is contained in${\displaystyle C^{\ast \ast }}$. (For some${\displaystyle C}$, namely the cones, the two are actually equal.)

The other two properties carry over without change:

• It is still true that an inclusion${\displaystyle C\subseteq D}$ is turned into an inclusion in the opposite direction (${\displaystyle D^{\ast }\subseteq C^{\ast }}$).
• Given two subsets${\displaystyle C}$ and${\displaystyle D}$ of the plane,${\displaystyle C}$ is contained in${\displaystyle D^{\ast }}$ if and only if${\displaystyle D}$ is contained in${\displaystyle C^{\ast }}$.

A very important example of a duality arises inlinear algebra by associating to anyvector spaceV itsdual vector spaceV^*. Its elements are thelinear functionals${\displaystyle \phi :V\to K}$, whereK is thefield over whichV is defined.The three properties of the dual cone carry over to this type of duality by replacing subsets of${\displaystyle {\mathbb{R}}^2}$ by vector space and inclusions of such subsets by linear maps. That is:

• Applying the operation of taking the dual vector space twice gives another vector spaceV^**. There is always a mapV →V^**. For someV, namely precisely thefinite-dimensional vector spaces, this map is anisomorphism.
• A linear mapV →W gives rise to a map in the opposite direction (W^* →V^*).
• Given two vector spacesV andW, the maps fromV toW^* correspond to the maps fromW toV^*.

A particular feature of this duality is thatV andV^* 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 abasis ofV. This is also true in the case ifV is aHilbert space,via theRiesz 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 a close relation between objects of seemingly different nature. One example of such a more general duality is fromGalois theory. For a fixedGalois extensionK /F, one may associate theGalois groupGal(K/E) to any intermediate fieldE (i.e.,F ⊆E ⊆K). This group is a subgroup of the Galois groupG = Gal(K/F). Conversely, to any such subgroupH ⊆G there is the fixed fieldK^H consisting of elements fixed by the elements inH.

Compared to the above, this duality has the following features:

• An extensionF ⊆F′ of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction:Gal(K/F′) ⊆ Gal(K/F).
• AssociatingGal(K/E) toE andK^H toH are inverse to each other. This is the content of thefundamental theorem of Galois theory.

Given aposetP = (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), thedual posetP^d = (X, ≥) comprises the same ground set but theconverse relation. Familiar examples of dual partial orders include

• the subset and superset relations⊂ and⊃ on any collection of sets, such as the subsets of a fixed setS. This gives rise to the first example of a duality mentionedabove.
• thedivides andmultiple-of relations on theintegers.
• thedescendant-of andancestor-of relations on the set of humans.

Aduality transform is aninvolutive antiautomorphismf of apartially ordered setS, that is, anorder-reversing involutionf :S →S.^[9][10] In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, iff₁,f₂ are two duality transforms then theircomposition is anorder automorphism ofS; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of apower setS = 2^R are induced by permutations ofR.

A concept defined for a partial orderP will correspond to adual concept on the dual posetP^d. For instance, aminimal element ofP will be amaximal element ofP^d: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts areupper and lower bounds,lower sets andupper sets, andideals andfilters.

In topology,open sets andclosed sets are dual concepts: the complement of an open set is closed, and vice versa. Inmatroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called thedual 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 the objects. A classical example (known to Euclid) of this is the duality of thePlatonic 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. Thedual polyhedron of any of these polyhedra may be formed as theconvex hull of the center points of each face of the primal polyhedron, so thevertices 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 other, so do the corresponding two parts of thedual polyhedron. More generally, using the concept ofpolar reciprocation, anyconvex polyhedron, or more generally anyconvex polytope, corresponds to adual polyhedron or dual polytope, with ani-dimensional feature of ann-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 theface lattices of the primal and dual polyhedra or polytopes are themselvesorder-theoretic duals. Duality of polytopes and order-theoretic duality are bothinvolutions: 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 aplanar graph, the graph of its vertices and edges. The dual polyhedron has adual 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 tograph embeddings 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 fromcomputational geometry: the duality for any finite setS of points in the plane between theDelaunay triangulation ofS and theVoronoi diagram ofS. 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.Matroid duality 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.

A kind of geometric duality also occurs inoptimization theory, but not one that reverses dimensions. Alinear program may be specified by a system of real variables (the coordinates for a point in Euclidean space${\displaystyle {\mathbb{R}}^n}$), a system of linear constraints (specifying that the point lie in ahalfspace; 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 adual 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 relationsA andB are considered dual ifA(¬x) = ¬B(x), where ¬ islogical 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 predicatesP in classical logic: if there exists anx for whichP fails to hold, then it is false thatP holds for allx (but the converse does not hold constructively). From this fundamental logical duality follow several others:

• A formula is said to besatisfiable in a certain model if there are assignments to itsfree variables that render it true; it isvalid ifevery 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 the quantifiers ranging over interpretations.
• In classical logic, the∧ and∨ operators are dual in this sense, because(¬x ∧ ¬y) and¬(x ∨y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem.De Morgan's laws are examples. More generally,∧ (¬x_i) = ¬∨x_i. The left side is true if and only if∀i.¬x_i, and the right side if and only if ¬∃i.x_i.
• Inmodal logic,□p means that the propositionp is "necessarily" true, and◊p thatp is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example inKripke semantics, "p is possibly true" means "there exists some worldW such thatp is true inW", while "p is necessarily true" means "for all worldsW,p is true inW". The duality of□ and◊ then follows from the analogous duality of∀ and∃. Other dual modal operators behave similarly. For example,temporal logic 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.

Other analogous dualities follow from these:

• Set-theoretic union and intersection are dual under theset complement operator^∁. That is,A^∁ ∩B^∁ = (A ∪B)^∁, and more generally,⋂A_α^∁ = (⋃A_α)^∁. This follows from the duality of∀ and∃: an elementx is a member of⋂A_α^∁ if and only if∀α.¬x ∈A_α, and is a member of(⋂A_α)^∁ if and only if¬∃α.x ∈A_α.

The dual of the dual, called thebidual ordouble dual, depending on context, is often identical to the original (also calledprimal), 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, 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 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 theidentity functor to the double dual functor. For vector spaces (considered algebraically), this is always an injection; seeDual space § Injection into the double-dual. This can be generalized algebraically to adual module. There is still a canonical evaluation map, but it is not always injective; if it is, this is known as atorsionless module; if it is an isomophism, the module is called reflexive.

Fortopological vector spaces (includingnormed vector spaces), there is a separate notion of atopological dual, denoted⁠${\displaystyle V^{\prime }}$⁠ to distinguish from the algebraic dualV^*, with different possible topologies on the dual, each of which defines a different bidual space⁠${\displaystyle V^{″}}$⁠. In these cases the canonical evaluation map⁠${\displaystyle V\to V^{″}}$⁠ is not in general an isomorphism. If it is, this is known (for certainlocally convex vector spaces with thestrong dual space topology) as areflexive space.

In other cases, showing a relation between the primal and bidual is a significant result, as inPontryagin duality (alocally compact abelian group is naturally isomorphic to its bidual).

A group of dualities can be described by endowing, for any mathematical objectX, the set of morphismsHom (X,D) into some fixed objectD, with a structure similar to that ofX. This is sometimes calledinternal Hom. In general, this yields a true duality only for specific choices ofD, in which caseX^* = Hom (X,D) is referred to as thedual ofX. There is always a map fromX to thebidual, that is to say, the dual of the dual,$${\displaystyle X\to X^{\ast \ast }:=(X^{\ast }{)}^{\ast }=\mathrm{Hom}(\mathrm{Hom}(X,D),D).}$$It assigns to somex ∈X the map that associates to any mapf :X →D (i.e., an element inHom(X,D)) the valuef(x).Depending on the concrete duality considered and also depending on the objectX, this map may or may not be an isomorphism.

The construction of the dual vector space$${\displaystyle V^{\ast }=\mathrm{Hom}(V,K)}$$mentioned in 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 mapV →V^** mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if thedimension ofV is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.

A vector spaceV is isomorphic toV^∗ precisely ifV is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degeneratebilinear form$${\displaystyle \phi :V\times V\to K}$$In this caseV is called aninner product space.For example, ifK is the field ofreal orcomplex numbers, anypositive definite bilinear form gives rise to such an isomorphism. InRiemannian geometry,V is taken to be thetangent space of amanifold and such positive bilinear forms are calledRiemannian 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 theHodge star which provides a correspondence between the elements of theexterior algebra. For ann-dimensional vector space, the Hodge star operator mapsk-forms to(n −k)-forms. This can be used to formulateMaxwell's equations. In this guise, the duality inherent in the inner product space exchanges the role ofmagnetic andelectric fields.

In someprojective planes, it is possible to findgeometric transformations 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.^[11] For such planes there arises a general principle ofduality 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.^[12] 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, theintersection point of these two lines". For further examples, seeDual 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${\displaystyle {\mathbb{R}\mathbb{P}}^2}$ correspond to one-dimensional subvector spaces${\displaystyle V\subset {\mathbb{R}}^3}$^[13] while the lines in the projective plane correspond to subvector spaces${\displaystyle W}$ of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional${\displaystyle V}$ the subspace of${\displaystyle ({\mathbb{R}}^3{)}^{\ast }}$ consisting of those linear maps${\displaystyle f:{\mathbb{R}}^3\to \mathbb{R}}$ which satisfy${\displaystyle f(V)=0}$. As a consequence of thedimension formula oflinear algebra, this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to${\displaystyle ({\mathbb{R}}^3{)}^{\ast }}$.

The (positive definite) bilinear form$${\displaystyle ⟨\cdot ,\cdot ⟩:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R},⟨x,y⟩=\sum _{i=1}^3{x}_i{y}_i}$$yields an identification of this projective plane with the${\displaystyle {\mathbb{R}\mathbb{P}}^2}$. Concretely, the duality assigns to${\displaystyle V\subset {\mathbb{R}}^3}$ itsorthogonal${\displaystyle \left\{w\in {\mathbb{R}}^3,⟨v,w⟩=0\text{ for all }v\in V\right\}}$. The explicit formulas induality in projective geometry arise by means of this identification.

In the realm oftopological vector spaces, a similar construction exists, replacing the dual by thetopological 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${\displaystyle X}$ that is canonically isomorphic to its bidual${\displaystyle X^{″}}$ is called areflexive space:$${\displaystyle X\cong X^{″}.}$$

Examples:

• As in the finite-dimensional case, on eachHilbert spaceH itsinner product⟨⋅, ⋅⟩ defines a map$${\displaystyle H\to H^{\ast },v\mapsto (w\mapsto ⟨w,v⟩),}$$ which is abijection due to theRiesz representation theorem. As a corollary, every Hilbert space is areflexive Banach space.
• Thedual normed space of anL^p-space isL^q where1/p + 1/q = 1 provided that1 ≤p < ∞, but the dual ofL^∞ is bigger thanL¹. HenceL¹ is not reflexive.
• Distributions are linear functionals on appropriate spaces of functions. They are an important technical means in the theory ofpartial differential equations (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.^[14] All the standard spaces of distributions —${\displaystyle {\mathcal{D}}^{\prime }(U)}$,${\displaystyle {\mathcal{S}}^{\prime }({\mathbb{R}}^n)}$,${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(U{)}^{\prime }}$ — are reflexive locally convex spaces.^[15]

Thedual lattice of alatticeL is given by$${\displaystyle \mathrm{Hom}(L,\mathbb{Z}),}$$the set of linear functions on thereal vector space containing the lattice that map the points of the lattice to the integers${\displaystyle \mathbb{Z}}$. This is used in the construction oftoric varieties.^[16] ThePontryagin dual oflocally compacttopological groupsG is given by$${\displaystyle \mathrm{Hom}(G,S^1),}$$continuousgroup homomorphisms with values in 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 ofcategory theory, this amounts to acontravariant functor between twocategoriesC andD:

which for any two objectsX andY ofC gives a map

That functor may or may not be anequivalence of categories. There are various situations, where such a functor is an equivalence between theopposite categoryC^op ofC, andD. 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.^[17] Therefore, any duality between categoriesC andD is formally the same as an equivalence betweenC andD^op (C^op andD). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.^[18]

A category that is equivalent to its dual is calledself-dual. An example of self-dual category is the category ofHilbert spaces.^[19]

Manycategory-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example,Cartesian productsY₁ ×Y₂ anddisjoint unionsY₁ ⊔Y₂ of sets are dual to each other in the sense that

and

for any setX. This is a particular case of a more general duality phenomenon, under whichlimits in a categoryC correspond tocolimits in the opposite categoryC^op; further concrete examples of this areepimorphisms vs.monomorphism, in particularfactor modules (or groups etc.) vs.submodules,direct products vs.direct sums (also calledcoproducts 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 areprojective andinjective modules inhomological algebra,^[20]fibrations andcofibrations in topology and more generallymodel categories.^[21]

TwofunctorsF:C →D andG:D →C areadjoint if for all objectsc inC andd inD

in 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 inC indexed by some categoryI its colimit and the diagonal functor that maps any objectc ofC to the constant diagram which hasc at all places. Dually,

Gelfand duality is a duality between commutativeC*-algebrasA andcompactHausdorff spacesX is the same: it assigns toX the space of continuous functions (which vanish at infinity) fromX toC, the complex numbers. Conversely, the spaceX can be reconstructed fromA as thespectrum ofA. Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way.^[22]

In a similar vein there is a duality inalgebraic geometry betweencommutative rings andaffine schemes: to every commutative ringA there is an affine spectrum,SpecA. Conversely, given an affine schemeS, one gets back a ring by taking global sections of thestructure sheaf O_S. In addition,ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence

(Commutative rings)^op ≅ (affine schemes)^[23]

Affine schemes are the local building blocks ofschemes. The previous result therefore tells that the local theory of schemes is the same ascommutative algebra, the study of commutative rings.

Noncommutative geometry draws inspiration from Gelfand duality and studies noncommutative C*-algebras as if they were functions on some imagined space.Tannaka–Krein duality is a non-commutative analogue of Pontryagin duality.^[24]

In a number of situations, the two categories which are dual to each other are actually arising frompartially ordered 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 aGalois connection. An example is the standard duality inGalois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extensionL ⊃K (inside some fixed bigger field Ω) the Galois group Gal (Ω /L) —to a smaller group.^[25]

The collection of all open subsets of a topological spaceX forms a completeHeyting algebra. There is a duality, known asStone duality, connectingsober spaces and spatiallocales.

• Birkhoff's representation theorem relatingdistributive lattices andpartial orders

Pontryagin duality gives a duality on the category oflocally compactabelian groups: given any such groupG, thecharacter group

χ(G) = Hom (G,S¹)

given by continuous group homomorphisms fromG to thecircle groupS¹ can be endowed with thecompact-open topology. Pontryagin duality states that the character group is again locally compact abelian and that

G ≅χ(χ(G)).^[26]

Moreover,discrete groups correspond tocompact abelian groups; 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 ofFourier analysis, see below.

Inanalysis, problems are frequently solved by passing to the dual description of functions and operators.

Fourier transform switches between functions on a vector space and its dual:$${\displaystyle \hat{f}(\xi ):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi ix\xi }dx,}$$and conversely$${\displaystyle f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{2\pi ix\xi }d\xi .}$$Iff is anL²-function onR orR^N, say, then so is${\displaystyle \hat{f}}$ and${\displaystyle f(-x)=\hat{\hat{f}}(x)}$. Moreover, the transform interchanges operations of multiplication andconvolution on the correspondingfunction spaces. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groupsR (orR^N etc.): any character ofR is given byξ ↦e^−2πixξ. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions ofquantum mechanical systems in terms of coordinate and momentum representations.

• Laplace transform is similar to Fourier transform and interchangesoperators of multiplication by polynomials with constant coefficientlinear differential operators.
• Legendre transformation is an important analytic duality which switches betweenvelocities inLagrangian mechanics andmomenta inHamiltonian mechanics.

Theorems showing that certain objects of interest are thedual spaces (in the sense of linear algebra) of other objects of interest are often calleddualities. Many of these dualities are given by abilinear pairing of twoK-vector spaces

A ⊗B →K.

Forperfect pairings, there is, therefore, an isomorphism ofA to thedual ofB.

Poincaré duality of a smooth compactcomplex manifoldX is given by a pairing of singular cohomology withC-coefficients (equivalently,sheaf cohomology of theconstant sheafC)

Hⁱ(X) ⊗ H²ⁿ⁻ⁱ(X) →C,

wheren is the (complex) dimension ofX.^[27] Poincaré duality can also be expressed as a relation ofsingular homology andde Rham cohomology, by asserting that the map

${\displaystyle (\gamma ,\omega )\mapsto {\int }_{\gamma }\omega }$

(integrating a differentialk-form over a (2n − k)-(real-)dimensional cycle) is a perfect pairing.

Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topologicalmanifold is represented as acell complex, 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 thekthhomology group and the (n − k)thcohomology group.

The same duality pattern holds for a smoothprojective variety over aseparably closed field, usingl-adic cohomology withQ_ℓ-coefficients instead.^[28] This is further generalized to possiblysingular varieties, usingintersection cohomology instead, a duality calledVerdier duality.^[29]Serre duality orcoherent duality are similar to the statements above, but applies to cohomology ofcoherent sheaves instead.^[30]

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 usingderived categories and certaindirect and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and theétale topology in the second case, and with respect to coherent sheaves for coherent duality).

Yet another group of similar duality statements is encountered inarithmetics: étale cohomology offinite,local andglobal fields (also known asGalois cohomology, since étale cohomology over a field is equivalent togroup cohomology of the (absolute)Galois group of the field) admit similar pairings. The absolute Galois groupG(F_q) of a finite field, for example, is isomorphic to${\displaystyle \hat{\mathbf{Z}}}$, theprofinite completion ofZ, the integers. Therefore, the perfect pairing (for anyG-moduleM)

Hⁿ(G,M) × H¹⁻ⁿ (G, Hom (M,Q/Z)) →Q/Z^[31]

is a direct consequence ofPontryagin duality of finite groups. For local and global fields, similar statements exist (local duality and global orPoitou–Tate duality).^[32]

• Atiyah, Michael (2007)."Duality in Mathematics and Physics lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB)"(PDF).
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• Gowers, Timothy (2008), "III.19 Duality",The Princeton Companion to Mathematics, Princeton University Press, pp. 187–190.
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