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Mathematics

Mathematics 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 space

Vector space

A 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 functional

Linear functional

In 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 finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra

Tensor algebra

In 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 infinite-dimensional), dual spaces are employed for defining and studying concepts like measures

Measure (mathematics)

In 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...

, distributions

Distribution (mathematics)

In 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 space

Hilbert space

The 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 two-dimensional Euclidean plane and three-dimensional 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 analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related 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 space

Topological vector space

In 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 space

Vector space

V over a field

Field (mathematics)

In 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 functional

Linear functional

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 one-form
One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree 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 .

Finite-dimensional case

If V is finite-dimensional, 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 R², and its basis chosen to be }, then e¹ and e² are one-form

One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree 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 Rⁿ as the space of columns of n real number

Real number

In 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 Rⁿ as a linear functional by ordinary matrix multiplication

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p 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.

Infinite-dimensional case

If V is not finite-dimensional but has a basis

Basis (linear algebra)

In 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 finite-dimensional 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 sequence

Sequence

In 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 non-zero 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_nx_n, which is a finite sum because there are only finitely many nonzero x_n. The dimension

Dimension (vector space)

In 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 infinite-dimensional 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 sum

Direct sum of modules

In 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 1-dimensional 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 product

Direct product

In 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 infinite-dimensional.

Bilinear products and dual spaces

If V is finite-dimensional, 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 finite-dimensional, 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 one-to-one 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 form

Sesquilinear form

In 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 conjugate

Complex conjugate vector space

In 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 complex-valued functionals such that

Injection into the double-dual

There is a natural

Natural transformation

In 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 isomorphism

Isomorphism

In 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 finite-dimensional. (Infinite-dimensional 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 isomorphism

Isomorphism

if and only if W is finite-dimensional. If then the space of linear maps is actually an algebra

Algebra over a field

In 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 antihomomorphism

Antihomomorphism

In 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 theory

Category theory

Category 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 matrix

Matrix (mathematics)

In 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 transpose

Transpose

In 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 Rⁿ, 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 annihilator
Annihilator (ring theory)
In 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 R-module. 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 finite-dimensional. 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 finite-dimensional, 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 connection

Galois connection

In 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 finite-dimensional vector space.

If W is a subspace of V then the quotient space

Quotient space (linear algebra)

In 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 kernel

Kernel (mathematics)

In 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 sum

Direct sum of modules

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 space

Topological vector space

In 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 finite-dimensional normed vector space or topological vector space, such as Euclidean n-space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional 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 infinite-dimensional normed space, as shown by the example of discontinuous linear map

Discontinuous linear map

In 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 space

Normed vector space

In mathematics, with 2- or 3-dimensional vectors with real-valued 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 space

Banach space

In 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 space

Hilbert space

) 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 topology

Weak topology

In 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

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

of all sequence

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 n-th 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 inequality

Hölder's inequality

In 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 convergent

Limit of a sequence

The 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 c₀ (the sequences converging to zero) are both naturally identified with .

By the Riesz representation theorem

Riesz representation theorem

There are several well-known 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 anti-isomorphic

Antiisomorphic

In 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 bra-ket notation

Bra-ket notation

Bra-ket 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 mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like 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 theorem

Hahn–Banach theorem

In 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 compact

Compact operator

In 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 adjoint

Hermitian adjoint

In 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 infinite-dimensional 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 topology

Strong topology (polar topology)

In 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 theorem

Hahn–Banach theorem

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 theorem

Hahn–Banach theorem

, this map is in fact an isometry

Isometry

In mathematics, an isometry is a distance-preserving 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 bijection

Bijection

A 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 reflexive

Reflexive space

In 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 space

Topological vector space

In 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 convex

Locally convex topological vector space

In 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 Hausdorff

Hausdorff space

In 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.

Algebraic dual space

Finite-dimensional case

Infinite-dimensional case

Bilinear products and dual spaces

Injection into the double-dual

Transpose of a linear map

Quotient spaces and annihilators

Continuous dual space

Examples

Transpose of a continuous linear map

Annihilators

Further properties

Double dual

See also