In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, any
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
,
V, has a corresponding
dual vector space (or just
dual space for short) consisting of all
linear functionalIn linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s on
V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in
tensor algebraIn mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
. When applied to vector spaces of functions (which typically are infinitedimensional), dual spaces are employed for defining and studying concepts like
measuresIn mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
,
distributionsIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
, and
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
s. Consequently, the dual space is an important concept in the study of
functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
.
There are two types of dual spaces: the
algebraic dual space, and the
continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a
topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
Algebraic dual space
Given any
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V over a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F, the
dual space V* is defined as the set of all linear maps (
linear functionalIn linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s). The dual space
V* itself becomes a vector space over
F when equipped with the following addition and scalar multiplication:

for all φ, ψ ∈ V*, x ∈ V, and a ∈ F. Elements of the algebraic dual space V* are sometimes called covectors or oneformIn linear algebra, a oneform on a vector space is the same as a linear functional on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s.
The pairing of a functional φ in the dual space V* and an element x of V is sometimes denoted by a bracket:
or to match the way the operation is performed i.e. [take a number, and perform this operation on it].
The pairing defines a nondegenerate bilinear mapping .
Finitedimensional case
If V is finitedimensional, then V* has the same dimension as V. Given a basis } in V, it is possible to construct a specific basis in V*, called the dual basis. This dual basis is a set } of linear functionals on V, defined by the relation

for any choice of coefficients . In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations

where δ_{ij} is the Kronecker delta symbol. For example if V is R^{2}, and its basis chosen to be }, then e^{1} and e^{2} are oneformIn linear algebra, a oneform on a vector space is the same as a linear functional on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s (functions which map a vector to a scalar) such that , , , and . (Note: The superscript here is the index, not an exponent).
In particular, if we interpret R^{n} as the space of columns of n real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s, its dual space is typically written as the space of rows of n real numbers. Such a row acts on R^{n} as a linear functional by ordinary matrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an nbym matrix and B is an mbyp matrix, the result AB of their multiplication is an nbyp matrix defined only if the number of columns m of the left matrix A is the...
.
If V consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of V* form a family of parallel lines in V. So an element of V* can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, one needs only to determine which of the lines the vector lies on. Or, informally, one "counts" how many lines the vector crosses. More generally, if V is a vector space of any dimension, then the level sets of a linear functional in V* are parallel hyperplanes in V, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.
Infinitedimensional case
If V is not finitedimensional but has a basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
e_{α} indexed by an infinite set A, then the same construction as in the finitedimensional case yields linearly independent elements e^{α} of the dual space, but they will not form a basis.
Consider, for instance, the space R^{∞}, whose elements are those sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of real numbers which have only finitely many nonzero entries, which has a basis indexed by the natural numbers N: for , e_{i} is the sequence which is zero apart from the ith term, which is one. The dual space of R^{∞} is R^{N}, the space of all sequences of real numbers: such a sequence (a_{n}) is applied to an element (x_{n}) of R^{∞} to give the number ∑a_{n}x_{n}, which is a finite sum because there are only finitely many nonzero x_{n}. The dimensionIn mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
of R^{∞} is countably infinite, whereas R^{N} does not have a countable basis.
This observation generalizes to any infinitedimensional vector space V over any field F: a choice of basis } identifies V with the space (F^{A})_{0} of functions such that is nonzero for only finitely many , where such a function ƒ is identified with the vector

in V (the sum is finite by the assumption on ƒ, and any may be written in this way by the definition of the basis).
The dual space of V may then be identified with the space F^{A} of all functions from A to F: a linear functional T on V is uniquely determined by the values it takes on the basis of V, and any function (with ) defines a linear functional T on V by

Again the sum is finite because ƒ_{α} is nonzero for only finitely many α.
Note that (F^{A})_{0} may be identified (essentially by definition) with the direct sumIn abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of infinitely many copies of F (viewed as a 1dimensional vector space over itself) indexed by A, i.e., there are linear isomorphisms

On the other hand F^{A} is (again by definition), the direct productIn mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of infinitely many copies of F indexed by A, and so the identification

is a special case of a general result relating direct sums (of modules) to direct products.
Thus if the basis is infinite, then there are always more vectors in the dual space than in the original vector space. This is in marked contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinitedimensional.
Bilinear products and dual spaces
If V is finitedimensional, then V is isomorphic to V*. But there is in general no natural isomorphism between these two spaces . Any bilinear form ⟨•,•⟩ on V gives a mapping of V into its dual space via
where the right hand side is defined as the functional on V taking each to <v,w>. In other words, the bilinear form determines a linear mapping
defined by
If the bilinear form is assumed to be nondegenerate, then this is an isomorphism onto a subspace of V*. If V is finitedimensional, then this is an isomorphism onto all of V*. Conversely, any isomorphism Φ from V to a subspace of V* (resp., all of V*) defines a unique nondegenerate bilinear form ⟨•,•⟩_{Φ} on V by
Thus there is a onetoone correspondence between isomorphisms of V to subspaces of (resp., all of) V* and nondegenerate bilinear forms on V.
If the vector space V is over the complex field, then sometimes it is more natural to consider sesquilinear formIn mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui meaning "one and a half"...
s instead of bilinear forms. In that case, a given sesquilinear form ⟨•,•⟩ determines an isomorphism of V with the complex conjugateIn mathematics, the complex conjugate of a complex vector space V\, is the complex vector space \overline V consisting of all formal complex conjugates of elements of V\,...
of the dual space

The conjugate space V* can be identified with the set of all additive complexvalued functionals such that

Injection into the doubledual
There is a naturalIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
homomorphism Ψ from V into the double dual V**, defined by for all , . This map Ψ is always injective; it is an isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
if and only if V is finitedimensional. (Infinitedimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.)
Transpose of a linear map
If is a linear map, then the transpose (or dual) is defined by

for every . The resulting functional ƒ*(φ) is in V*, and is called as the pullback of φ along ƒ.
The following identity holds for all and :

where the bracket [•,•] on the left is the duality pairing of V with its dual space, and that on the right is the duality pairing of W with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint.
The assignment produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
if and only if W is finitedimensional. If then the space of linear maps is actually an algebraIn mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
under composition of maps, and the assignment is then an antihomomorphismIn mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection from an object to...
of algebras, meaning that . In the language of category theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. Note that one can identify (ƒ*)* with ƒ using the natural injection into the double dual.
If the linear map ƒ is represented by the matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
A with respect to two bases of V and W, then ƒ* is represented by the transposeIn linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
matrix A^{T} with respect to the dual bases of W* and V*, hence the name. Alternatively, as ƒ is represented by A acting on the left on column vectors, ƒ* is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on R^{n}, which identifies the space of column vectors with the dual space of row vectors.
Quotient spaces and annihilators
Let S be a subset of V. The annihilatorIn mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.Definitions:Let R be a ring, and let M be a left Rmodule. Choose a nonempty subset S of M...
of S in V*, denoted here S^{o}, is the collection of linear functionals such that for all . That is, S^{o} consists of all linear functionals such that the restriction to S vanishes: .
The annihilator of a subset is itself a vector space. In particular, is all of V* (vacuously), whereas is the zero subspace. Furthermore, the assignment of an annihilator to a subset of V reverses inclusions, so that if , then

Moreover, if A and B are two subsets of V, then

and equality holds provided V is finitedimensional. If A_{i} is any family of subsets of V indexed by i belonging to some index set I, then

In particular if A and B are subspaces of V, it follows that

If V is finitedimensional, and W is a vector subspace, then

after identifying W with its image in the second dual space under the double duality isomorphism . Thus, in particular, forming the annihilator is a Galois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
on the lattice of subsets of a finitedimensional vector space.
If W is a subspace of V then the quotient spaceIn linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
V/W is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional factors through V/W if and only if W is in the kernelIn mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
of ƒ. There is thus an isomorphism

As a particular consequence, if V is a direct sumIn abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of two subspaces A and B, then V* is a direct sum of A^{o} and B^{o}.
Continuous dual space
When dealing with topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s, one is typically only interested in the continuous linear functionals from the space into the base field. This gives rise to the notion of the"continuous dual space" which is a linear subspace of the algebraic dual space V*, denoted . For any finitedimensional normed vector space or topological vector space, such as Euclidean nspaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, the continuous dual and the algebraic dual coincide. This is however false for any infinitedimensional normed space, as shown by the example of discontinuous linear mapIn mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions . If the spaces involved are also topological spaces , then it makes sense to ask whether all linear maps...
s.
The continuous dual of a normed vector spaceIn mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
V (e.g., a Banach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
or a Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
) forms a normed vector space. A norm φ of a continuous linear functional on V is defined by

This turns the continuous dual into a normed vector space, indeed into a Banach space so long as the underlying field is complete, which is often included in the definition of the normed vector space. In other words, this dual of a normed space over a complete field is necessarily complete.
The continuous dual can be used to define a new topology on V, called the weak topologyIn mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
.
Examples
Let 1 < p < ∞ be a real number and consider the Banach space ℓ^{ p}In mathematics, the Lp spaces are function spaces defined using a natural generalization of the pnorm for finitedimensional vector spaces...
of all sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s for which

is finite. Define the number q by . Then the continuous dual of ℓ^{ p} is naturally identified with ℓ^{ q}: given an element , the corresponding element of is the sequence (φ(e_{n})) where e_{n} denotes the sequence whose nth term is 1 and all others are zero. Conversely, given an element , the corresponding continuous linear functional φ on is defined by for all (see Hölder's inequalityIn mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
).
In a similar manner, the continuous dual of is naturally identified with (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergentThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
sequences, with the supremum norm) and c_{0} (the sequences converging to zero) are both naturally identified with .
By the Riesz representation theoremThere are several wellknown theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. The Hilbert space representation theorem :...
, the continuous dual of a Hilbert space is again a Hilbert space which is antiisomorphicIn modern algebra, an antiisomorphism between structured sets A and B is an isomorphism from A to the opposite of B...
to the original space. This gives rise to the braket notationBraket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
used by physicists in the mathematical formulation of quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Transpose of a continuous linear map
If is a continuous linear map between two topological vector spaces, then the (continuous) transpose is defined by the same formula as before:

The resulting functional is in. The assignment produces a linear map between the space of continuous linear maps from V to W and the space of linear maps from to . When T and U are composable continuous linear maps, then

When V and W are normed spaces, the norm of the transpose in is equal to that of T in. Several properties of transposition depend upon the Hahn–Banach theoremIn mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
. For example, the bounded linear map T has dense range if and only if the transpose is injective.
When T is a compactIn functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
linear map between two Banach spaces V and W, then the transpose is compact. This can be proved using the Arzelà–Ascoli theorem.
When V is a Hilbert space, there is an antilinear isomorphism i_{V} from V onto its continuous dual. For every bounded linear map T on V, the transpose and the adjointIn mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinitedimensional situations...
operators are linked by

When T is a continuous linear map between two topological vector spaces V and W, then the transpose is continuous when and are equipped with"compatible" topologies: for example when, for and , both duals have the strong topologyIn functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair...
of uniform convergence on bounded sets of X, or both have the weak∗ topology of pointwise convergence on X. The transpose is continuous from to , or from to .
Annihilators
Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in,

Then, the dual of the quotient can be identified with W^{⊥}, and the dual of W can be identified with the quotient . Indeed, let P denote the canonical surjection from V onto the quotient ; then, the transpose is an isometric isomorphism from into, with range equal to W^{⊥}. If j denotes the injection map from W into V, then the kernel of the transpose is the annihilator of W:
and it follows from the Hahn–Banach theoremIn mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
that induces an isometric isomorphism
.
Further properties
If the dual of a normed space V is separable, then so is the space V itself. The converse is not true: for example the space is separable, but its dual is not.
Double dual
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator from a normed space V into its continuous double dual, defined by

As a consequence of the Hahn–Banach theoremIn mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
, this map is in fact an isometryIn mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, meaning for all x in V. Normed spaces for which the map Ψ is a bijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
are called reflexiveIn functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties. Normed spaces :Suppose X is a normed vector space over R or C...
.
When V is a topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
, one can still define Ψ(x) by the same formula, for every , however several difficulties arise. First, when V is not locally convexIn functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is HausdorffIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and locally convex, the map Ψ is injective from V to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.
Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual , so that the continuous double dual is not uniquely defined as a set. Saying that Ψ maps from V to , or in other words, that Ψ(x) is continuous on for every , is a reasonable minimal requirement on the topology of , namely that the evaluation mappings

be continuous for the chosen topology on . Further, there is still a choice of a topology on , and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.
See also
 Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a onetoone fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
 Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
 Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.Introduction:...
 Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly kspace, due to the relationship between the Pontryagin duals momentum and...
– dual space basis, in crystallography
 Covariance and contravariance of vectors