In

_{''i''}.
* In

_{''S''}. In addition, ^{op} ≅ (affine schemes)
Affine schemes are the local building blocks of schemes. The previous result therefore tells that the local theory of schemes is the same as

^{1})
given by continuous group homomorphisms from ''G'' to the ^{1} can be endowed with the

^{2}-function on R or R^{''N''}, say, then so is $\backslash widehat$ and $f(-x)\; =\; \backslash widehat(x)$. Moreover, the transform interchanges operations of multiplication and ^{''N''} etc.): any character of R is given by ξ ↦ ''e''^{−2''πixξ''}. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of

^{''i''}(X) ⊗ H^{2''n''−''i''}(X) → C,
where ''n'' is the (complex) dimension of ''X''. Poincaré duality can also be expressed as a relation of

_{ℓ}-coefficients instead. This is further generalized to possibly singular varieties, using intersection cohomology instead, a duality called Verdier duality. Serre duality or coherent duality are similar to the statements above, but applies to cohomology of _{''q''}) of a finite field, for example, is isomorphic to $\backslash widehat$, the ^{''n''}(''G'', ''M'') × H^{1−''n''} (''G'', Hom (''M'', Q/Z)) → Q/Z
is a direct consequence of Pontryagin duality of finite groups. For local and global fields, similar statements exist ( local duality and global or Poitou–Tate duality).;

Duality in Mathematics and Physics

lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB). *. *. * (a non-technical overview about several aspects of geometry, including dualities)

A History of Duality in Algebraic Topology

* . Als

* * * * * * * * * * * * * * * * * * {{cite book , last = Edwards , first = R. E. , year = 1965 , title = Functional analysis. Theory and applications , publisher = Holt, Rinehart and Winston , location = New York , isbn = 0030505356 * ja:双対

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a duality translates concepts, theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

s or mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additio ...

s into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...

''.
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 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 function
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...

s'' corresponds to the pairing in which one integrates a distribution against a test function, and '' Poincaré duality'' corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.
From a category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...

viewpoint, duality can also be seen as a functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

, 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 its dual .
Introductory examples

In the words of Michael Atiyah, 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.Complement of a subset

A simple, maybe the most simple, duality arises from consideringsubset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

s of a fixed set . To any subset , the complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...

consists of all those elements in that are not contained in . It is again a subset of . Taking the complement has the following properties:
* Applying it twice gives back the original set, i.e., . This is referred to by saying that the operation of taking the complement is an '' involution''.
* An inclusion of sets is turned into an inclusion in the ''opposite'' direction .
* Given two subsets and of , is contained in if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...

is contained in .
This duality appears in topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

as a duality between open and closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

s of some fixed topological space : a subset of is closed if and only if its complement in 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 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 is equal to the closure of the complement of .
Dual cone

A duality ingeometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

is provided by the dual cone construction. Given a set $C$ of points in the plane $\backslash mathbb\; R^2$ (or more generally points in the dual cone is defined as the set $C^*\; \backslash subseteq\; \backslash mathbb\; R^2$ consisting of those points $(x\_1,\; x\_2)$ satisfying
$$x\_1\; c\_1\; +\; x\_2\; c\_2\; \backslash ge\; 0$$
for all points $(c\_1,\; c\_2)$ in $C$, 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 $C$. Instead, $C^$ is the smallest cone containing $C$ which may be bigger than $C$. Therefore this duality is weaker than the one above, in that
* Applying the operation twice gives back a possibly bigger set: for all $C$, $C$ is contained in $C^$. (For some $C$, namely the cones, the two are actually equal.)
The other two properties carry over without change:
* It is still true that an inclusion $C\; \backslash subseteq\; D$ is turned into an inclusion in the opposite direction ($D^*\; \backslash subseteq\; C^*$).
* Given two subsets $C$ and $D$ of the plane, $C$ is contained in $D^*$ if and only if $D$ is contained in $C^*$.
Dual vector space

A very important example of a duality arises inlinear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...

by associating to any vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...

its dual vector space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

. Its elements are the linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , ...

s $\backslash varphi:\; V\; \backslash to\; K$, where is the field over which is defined.
The three properties of the dual cone carry over to this type of duality by replacing subsets of $\backslash mathbb\; R^2$ 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 space . There is always a map . For some , namely precisely the finite-dimensional vector space
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...

s, this map is an isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

.
* A linear map gives rise to a map in the opposite direction ().
* Given two vector spaces and , the maps from to correspond to the maps from to .
A particular feature of this duality is that and 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 basis of . This is also true in the case if is a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

, ''via'' the Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called th ...

.
Galois theory

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 from Galois theory. For a fixed Galois extension , one may associate the Galois group to any intermediate field (i.e., ). This group is a subgroup of the Galois group . Conversely, to any such subgroup there is the fixed field consisting of elements fixed by the elements in . Compared to the above, this duality has the following features: * An extension of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction: . * Associating to and to are inverse to each other. This is the content of the fundamental theorem of Galois theory.Order-reversing dualities

Given aposet
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

(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 comprises the same ground set but the converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...

. 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 set . This gives rise to the first example of a duality mentioned above.
* the ''divides'' and ''multiple-of'' relations on the integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

.
* the ''descendant-of'' and ''ancestor-of'' relations on the set of humans.
A ''duality transform'' is an involutive antiautomorphism of a partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

, that is, an order-reversing involution . In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if , are two duality transforms then their composition is an order automorphism of ; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

are induced by permutations of .
A concept defined for a partial order will correspond to a ''dual concept'' on the dual poset . For instance, a minimal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is def ...

of will be a maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defi ...

of : minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an ele ...

, lower set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...

s and upper set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...

s, and ideals and filters.
In topology, open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...

s and closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

s are dual concepts: the complement of an open set is closed, and vice versa. In matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being i ...

theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.
Dimension-reversing dualities

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 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. Thedual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the othe ...

of any of these polyhedra may be formed as the convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...

of the center points of each face of the primal polyhedron, so the vertices 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 the dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the othe ...

. More generally, using the concept of polar reciprocation, any convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

, or more generally any convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

, corresponds to a dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the othe ...

or dual polytope, with an -dimensional feature of an -dimensional polytope corresponding to an -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: 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
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

, the graph of its vertices and edges. The dual polyhedron has a dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loo ...

, 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 embedding
In topological graph theory, an embedding (also spelled imbedding) of a graph G on a surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with vertices and simple arcs ( homeomorphic images of ,1/math>) ...

s 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 computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ...

: the duality for any finite set of points in the plane between the Delaunay triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle ...

of and the Voronoi diagram of . 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 in optimization theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, 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 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.
Duality in logic and set theory

In logic, functions or relations and are considered dual if , where ¬ islogical negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...

. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because and are equivalent for all predicates in classical logic: if there exists an for which fails to hold, then it is false that holds for all (but the converse does not hold constructively). From this fundamental logical duality follow several others:
* A formula is said to be ''satisfiable
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over ...

'' in a certain model if there are assignments to its free variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is ...

s that render it true; it is ''valid'' if ''every'' 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 and are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathe ...

are examples. More generally, . The left side is true if and only if , and the right side if and only if ¬∃''i''.''x''modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

, means that the proposition is "necessarily" true, and that is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example in Kripke semantics, " is possibly true" means "there exists some world such that is true in ", while " is necessarily true" means "for all worlds , is true in ". The duality of and then follows from the analogous duality of and . Other dual modal operators behave similarly. For example, temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' ...

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 the set complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...

operator . That is, , and more generally, . This follows from the duality of and : an element is a member of if and only if , and is a member of if and only if .
Dual objects

A group of dualities can be described by endowing, for any mathematical object , the set of morphisms into some fixed object , with a structure similar to that of . This is sometimes called internal Hom. In general, this yields a true duality only for specific choices of , in which case is referred to as the ''dual'' of . There is always a map from to the ''bidual'', that is to say, the dual of the dual, $$X\; \backslash to\; X^\; :=\; (X^*)^*\; =\; \backslash operatorname(\backslash operatorname(X,\; D),\; D).$$ It assigns to some the map that associates to any map (i.e., an element in ) the value . Depending on the concrete duality considered and also depending on the object , this map may or may not be an isomorphism.Dual vector spaces revisited

The construction of the dual vector space $$V^*\; =\; \backslash operatorname(V,\; K)$$ mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e.,linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

s, forms a vector space in its own right. The map mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...

of is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.
Isomorphisms of and and inner product spaces

A vector space is isomorphic to precisely if is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degeneratebilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear ...

$$\backslash varphi:\; V\; \backslash times\; V\; \backslash to\; K$$
In this case is called an inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often de ...

.
For example, if is the field of real or complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, any positive definite bilinear form gives rise to such an isomorphism. In Riemannian geometry, is taken to be the tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

of a manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

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
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

. For an -dimensional vector space, the Hodge star operator maps -forms to -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
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particl ...

and electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

s.
Duality in projective geometry

In someprojective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

s, 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 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 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 $\backslash mathbb^2$ correspond to one-dimensional subvector spaces $V\; \backslash subset\; \backslash mathbb\; R^3$ while the lines in the projective plane correspond to subvector spaces $W$ of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional $V$ the subspace of $(\backslash mathbb\; R^3)^*$ consisting of those linear maps $f:\; \backslash mathbb\; R^3\; \backslash to\; \backslash mathbb\; R$ which satisfy $f\; (V)\; =\; 0$. As a consequence of the dimension formula of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...

, this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to $(\backslash mathbb\; R^3)^*$.
The (positive definite) bilinear form
$$\backslash langle\; \backslash cdot\; ,\; \backslash cdot\; \backslash rangle\; :\; \backslash R^3\; \backslash times\; \backslash R^3\; \backslash to\; \backslash R,\; \backslash langle\; x\; ,\; y\; \backslash rangle\; =\; \backslash sum\_^3\; x\_i\; y\_i$$
yields an identification of this projective plane with the $\backslash mathbb^2$. Concretely, the duality assigns to $V\; \backslash subset\; \backslash mathbb\; R^3$ its orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

$\backslash left\backslash $. The explicit formulas in duality in projective geometry arise by means of this identification.
Topological vector spaces and Hilbert spaces

In the realm oftopological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...

s, 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 $X$ that is canonically isomorphic to its bidual $X\text{'}\text{'}$ is called a reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an is ...

:
$$X\backslash cong\; X\text{'}\text{'}.$$
Examples:
* As in the finite-dimensional case, on each Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

its inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often de ...

defines a map $$H\; \backslash to\; H^*,\; v\; \backslash mapsto\; (w\; \backslash mapsto\; \backslash langle\; w,v\; \backslash rangle),$$ which is a bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

due to the Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called th ...

. As a corollary, every Hilbert space is a reflexive Banach space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an is ...

.
* The dual normed space of an -space is where provided that , but the dual of is bigger than . Hence is not reflexive.
* Distributions are linear functionals on appropriate spaces of functions. They are an important technical means in the theory of partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...

s (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 — $\text{'}(U)$, $\text{'}(\backslash R^n)$, $^\backslash infty(U)\text{'}$ — are reflexive locally convex spaces.
Further dual objects

The dual lattice of a lattice is given by $$\backslash operatorname\; (L,\; \backslash mathbf),$$ which is used in the construction of toric varieties. ThePontryagin dual
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...

of locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...

topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

s ''G'' is given by
$$\backslash operatorname\; (G,\; S^1),$$
continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation).
Dual categories

Opposite category and adjoint functors

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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...

, this amounts to a contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

between two categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...

and :
which for any two objects ''X'' and ''Y'' of ''C'' gives 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 of , and . 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 and is formally the same as an equivalence between and ( and ). 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 space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

s.
Many category-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example, Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...

s and disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...

s of sets are dual to each other in the sense that
and
for any set . This is a particular case of a more general duality phenomenon, under which limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...

in a category correspond to colimits in the opposite category ; further concrete examples of this are epimorphisms vs. monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphi ...

, in particular factor modules (or groups etc.) vs. submodules, direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...

s vs. direct sums (also called coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprod ...

s 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 projective and injective modules in homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...

, fibrations and cofibrations in topology and more generally model categories.
Two functors and are adjoint if for all objects ''c'' in ''C'' and ''d'' in ''D''
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 in indexed by some category its colimit and the diagonal functor that maps any object of to the constant diagram which has at all places. Dually,
Spaces and functions

Gelfand duality is a duality between commutativeC*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuo ...

s ''A'' and compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...

s ''X'' 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
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

between commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

s and affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...

s: to every commutative ring ''A'' there 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
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...

Oring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preserv ...

s are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence
: (Commutative rings)commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promine ...

, the study of commutative rings.
Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...

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.
Galois connections

In a number of situations, the two categories which are dual to each other are actually arising frompartially ordered
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

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 Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...

. An example is the standard duality in Galois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension ''L'' ⊃ ''K'' (inside some fixed bigger field Ω) the Galois group Gal (Ω / ''L'') —to a smaller group.
The collection of all open subsets of a topological space ''X'' forms a complete Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''i ...

. There is a duality, known as Stone duality, connecting sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definitio ...

s and spatial locales.
* Birkhoff's representation theorem relating distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...

s and partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

s
Pontryagin duality

Pontryagin duality gives a duality on the category oflocally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...

abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

s: given any such group ''G'', the character group
:χ(''G'') = Hom (''G'', ''S''circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \. ...

''S''compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory an ...

. Pontryagin duality states that the character group is again locally compact abelian and that
:''G'' ≅ χ(χ(''G'')).
Moreover, discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...

s correspond to compact 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 of Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

, see below.
Analytic dualities

Inanalysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...

, problems are frequently solved by passing to the dual description of functions and operators.
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

switches between functions on a vector space and its dual:
$$\backslash widehat(\backslash xi)\; :=\; \backslash int\_^\backslash infty\; f(x)\backslash \; e^\; \backslash ,\; dx,$$
and conversely
$$f(x)\; =\; \backslash int\_^\backslash infty\; \backslash widehat(\backslash xi)\backslash \; e^\; \backslash ,\; d\backslash xi.$$
If ''f'' is an ''L''convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

on the corresponding function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

s. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups R (or Rquantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

systems in terms of coordinate and momentum representations.
* Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the co ...

is similar to Fourier transform and interchanges operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...

of multiplication by polynomials with constant coefficient linear differential operators.
* Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...

is an important analytic duality which switches between velocities in Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-L ...

and momenta in Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''moment ...

.
Homology and cohomology

Theorems showing that certain objects of interest are thedual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...

s (in the sense of linear algebra) of other objects of interest are often called ''dualities''. Many of these dualities are given by a bilinear pairing of two ''K''-vector spaces
:''A'' ⊗ ''B'' → ''K''.
For perfect pairings, there is, therefore, an isomorphism of ''A'' to the dual of ''B''.
Poincaré duality

Poincaré duality of a smooth compactcomplex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a c ...

''X'' is given by a pairing of singular cohomology with C-coefficients (equivalently, sheaf cohomology of the constant sheaf C)
:Hsingular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...

and de Rham cohomology, by asserting that the map
:$(\backslash gamma,\; \backslash omega)\; \backslash mapsto\; \backslash int\_\backslash gamma\; \backslash omega$
(integrating a differential ''k''-form over an 2''n''−''k''-(real) -dimensional cycle) is a perfect pairing.
Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

is represented as a cell 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 the ''k''th homology group and the (''n'' − ''k'')th cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

group.
Duality in algebraic and arithmetic geometry

The same duality pattern holds for a smoothprojective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

over a separably closed field, using l-adic cohomology
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extens ...

with Qcoherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...

instead.
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 derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the étale topology in the second case, and with respect to coherent sheaves for coherent duality).
Yet another group of similar duality statements is encountered in arithmetics
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...

: étale cohomology of finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...

, local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administ ...

and global fields (also known as Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natu ...

, since étale cohomology over a field is equivalent to group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...

of the (absolute) Galois group of the field) admit similar pairings. The absolute Galois group ''G''(Fprofinite completion In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

of Z, the integers. Therefore, the perfect pairing (for any ''G''-module ''M'')
:HSee also

* Adjoint functor * Autonomous category *Dual abelian variety In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which is ...

* Dual basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...

* Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the s ...

* Dual code
* Duality (electrical engineering)
* Duality (optimization)
* Dualizing module
* Dualizing sheaf
* Dual lattice
* Dual norm
* Dual numbers, a certain associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...

; the term "dual" here is synonymous with ''double'', and is unrelated to the notions given above.
* Dual system
* Koszul duality
* Langlands dual
* Linear programming#Duality
* List of dualities
* Matlis duality
* Petrie duality
* Pontryagin duality
* S-duality
In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theor ...

* T-duality
In theoretical physics, T-duality (short for target-space duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories des ...

, Mirror symmetry
Notes

References

Duality in general

* Atiyah, Michael (2007)Duality in Mathematics and Physics

lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB). *. *. * (a non-technical overview about several aspects of geometry, including dualities)

Duality in algebraic topology

*James C. Becker and Daniel Henry GottliebA History of Duality in Algebraic Topology

Specific dualities

* . Als* . Als

* * * * * * * * * * * * * * * * * * {{cite book , last = Edwards , first = R. E. , year = 1965 , title = Functional analysis. Theory and applications , publisher = Holt, Rinehart and Winston , location = New York , isbn = 0030505356 * ja:双対