In mathematics it can be shown that any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. In many cases, these two spaces are isomorphic which means that the distinction between their elements is not always apparent. However, both in mathematics and applied sciences such as physics it is sometimes necessary to make a clear distinction between a vector space and its dual space to make the correct calculations. For example, if V is a space of points in a Euclidean 3D space, then a gradient of a function defined on these points is an element of the corresponding dual space. For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...

Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. When applied to vector spaces of functions (which typically are infinite dimensional) dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces. Consequently, dual space is an important concept in the study of functional analysis. For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

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. The continuous dual space is a particular type of algebraic dual space which is defined only for topological vector spaces. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

Algebraic dual space

Motivation (example)

Consider a function (this means that given the vector is a real number). The gradient of f with respect to , here denoted , is a vector in (this vector is a function of too) which gives the infinitesimal change in f, when moving the infinitesimal displacement , according to

df = ,

This means that the infinitesimal change in f is given by the scalar product between two vectors in . However, a closer examination of this expression reveals that the two vectors, in fact, are elements of two different vectors spaces. If we make a simple change of variable: , the following transformations take place in the two vectors which are scalar multiplied:

As a consequence, the infinitesimal change in f transformes as

df =

which is consistent with the fact that df is invariant with respect to changes of variables. However, we also note that the two vectors in , which are scalar multiplied to give df, transform in opposite ways. One is divided by 2 and one is multiplied by 2. If both vectors belong to the same vector space, they should transform in the same way. Since they do not, we conclude that they are elements of two distinct vector spaces, both isomorphic to .

The space which the gradient belongs to is the dual space relative to the space which or belong to. These two spaces are strongly connected, to any vector space V there is a corresponding dual space, denoted V ^* , which is formally defined below. The characteristic difference between elements of V and V ^* , is that coordinate transformations of vectors in V related to basis transformations correspond to opposite or dual transformations of coordinates related to vectors in V ^* .

Formal definition

Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on V, i.e., scalar-valued linear transformations on V (in this context, a "scalar" is a member of the base-field F). V* itself becomes a vector space over F under the following definition of addition and scalar multiplication: In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

for all φ, ψ in V*, a in F and x in V. In the language of tensors, elements of V are sometimes called covariant vectors, and elements of V*, contravariant vectors, covectors or one-forms. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

Double dual space

V ^* is a vector space and consequently it too has a dual space consisting of all linear functional on V ^* . This double dual space is often denoted V ^{* *} . In many cases, it is possible to show that V ^{* *} = V or that V ^{* *} is isomorphic to V. For example, this relation is satisfied if V is of finite dimension. However, for vectors spaces of infinite dimension this relation can be valid but does not have to be.

Dual basis

Given that V is of finite dimension and that the set {e₁,...,e_n} is a basis for V, then a corresponding dual basis in V* can be constructed as follows: In mathematics, the dimension of a vector space V is the cardinality (i. ...

Let {b¹,...,bⁿ} be an arbitrary basis for V*. Since the basis elements belong to V* they are linear functionals on V, which means that each of them can be applied to each of the basis vectors in V and the result is a set of scalars. These scalars can be collected into a matrix A:

The matrix A cannot be singular. If it was, there would exist a set of scalars c_j, not all equal to zero, such that:

for every i. This leads to the conclusion that

but since the vectors {e_j} constitute a basis this cannot be true. Consequently, A is not singular and therefore has a well-defined inverse . Now form a new set of vectors in V* in terms of linear combinations of the basis {b^k} with the elements of D:

First of all, we can safely say that {eⁱ} is a basis for V* since this set was produced by a non-singular linear transformation of the basis {bⁱ}. Second, this new basis in V* has a particular relation to the basis {e_j} in V:

This last property is the distinctive character of a dual basis, but note that {eⁱ} is not a dual basis in general, it is a dual basis relative to the basis {e_j} in V. If we choose another basis in V, then we get another dual basis in V*. The above derivation shows that a dual basis always exists, at least for the finite-dimensional case. Furthermore, in this case it is straightforward to show that the dual basis also is unique. However, for the infinite dimensional case existence and uniqueness may not be provable depending on the type of space V.

Example

In the case of R², its basis is B={e₁=(1,0),e₂=(0,1)}.Then, e¹ is a one-form (function which maps a vector to a scalar) such that e¹(e₁)=1, and e¹(e₂)=0. Similarity for e². (Note: The superscript here is an index, not an exponent.) If we give R² the traditional Cartesian coordinate system with x- and y-axes, e¹ gives the x-component of a point, and e² gives the y-component. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Concretely, if we interpret Rⁿ as space of columns of n real numbers, 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. In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... This article gives an overview of the various ways to multiply matrices. ...

If V consists of the space of geometrical vectors (arrows) in the plane, then the elements of the dual V* can be intuitively represented as collections of parallel lines. Such a collection of lines can be applied to a vector to yield a number in the following way: one counts how many of the lines the vector crosses. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

If V is infinite-dimensional, then the above construction of eⁱ does not produce a basis for V* and the dimension of V* is greater than that of V. Consider for instance the space R^(ω), whose elements are those sequences of real numbers which have only finitely many non-zero entries (dimension is countably infinite). The dual of this space is R^ω, the space of all sequences of real numbers (dimension is uncountably infinite). Such a sequence (a_n) is applied to an element (x_n) of R^(ω) to give the number ∑_na_nx_n. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

Bilinear products and dual spaces

As we saw above, if V is finite-dimensional, then V is isomorphic to V*, but the isomorphism is not natural and depends on the basis of V we started out with. In fact, any isomorphism Φ from V to V* defines a unique non-degenerate bilinear form on V by In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

and conversely every such non-degenerate bilinear product on a finite-dimensional space gives rise to an isomorphism from V to V*.

Injection into the double-dual

There is a natural homomorphism Ψ from V into the double dual V**, defined by (Ψ(v))(φ) = φ(v) for all v in V, φ in V*. This map Ψ is always injective; it is an isomorphism if and only if V is finite-dimensional. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

Pullback of a linear map

If f: V→W is a linear map, we may define its pullback f*: W*→V* by In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

where φ is an element of W*.

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 if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that (fg)*=g*f*. In the language of category theory, taking the dual of vector spaces and the pullback of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. Note that one can identify (f*)* with f using the natural injection into the double dual. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... For functors in computer science, see the function object article. ...

If the linear map f is represented by the matrix A with respect to two bases of V and W, then f* is represented by same matrix acting by multiplication on the right on row vectors. Using the canonical inner product on Rⁿ, one may identify the space with its dual, in which case the matrix can be represented by the transposed matrix ^tA. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

Structure of the dual space

The structure of the algebraic dual space is simply related to the structure of the vector space. If the space is finite dimensional then the space and its dual are isomorphic, while if the space is infinite dimensional then the dual space always has larger dimension.

Given a basis {e_α} for V indexed by A, one may construct the linearly independent set of dual vectors {σ^α}, as defined above. If V is infinite-dimensional however, the dual vectors do not form a basis for V*; the span of {σ^α} consists of all finite linear combinations of the dual vectors, but any infinite ordered tuple of dual vectors (thought of informally as an infinite sum) defines an element of the dual space. Because every vector of the vector space may be written as a finite linear combination of basis vectors {e_α}, an infinite tuple of dual vectors evaluates to nonzero scalars only finitely many times. In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...

More explicitly, any infinite tuple (f_ασ^α) may be thought of as the infinite sum

which satisfies

So f acts on an arbitrary vector

in V by

This dual vector f is linearly independent of the dual vectors {σ^α} unless A is finite. The dual space is the span of all such tuples. The idea of a dual vector as an infinite sum should not be taken too literally; in general infinite sums are defined in terms of a limit, which only makes sense in a topological space, and even then not all sums will be convergent. A basis for the dual space is a set of vectors such that every dual vector can be written as a finite linear combination. The existence of such a basis requires the axiom of choice, and cannot be exhibited explicitly. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

This can be understood more rigorously, if perhaps more abstractly, as follows: For any vector space V over F, we can find a basis. If that basis has cardinality α (thus α is the dimension of the vector space), then we may find a basis indexed by α. Since any field may be viewed as a one dimensional vector space over itself, we may construct the vector space direct sum of copies of F and the existence of the basis is equivalent to the existence of an isomorphism In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, an index set is another name for a function domain. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Thus this isomorphism is nothing other than the equivalent statement that any vector can be uniquely written as a sum of finitely many basis vectors, which is simply the definition of a basis. Note that the isomorphism is not canonical; it depends on the particular choice of basis.

A property of the direct sum is that the operation of passing to the dual turns direct sums into direct products. That is, In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...

and here in the second equation we use the fact that any field F, viewed as a vector space over itself, is canonically isomorphic to its dual space. Thus we see that

Recall that the vector space direct sum is the set of tuples which are only nonzero finitely many times, while the vector space direct product is the set of all tuples (tuples which may be nonzero infinitely often). If α is infinite, then there are always more vectors in the dual space than the vector space. This is in marked contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the vector space even for infinite-dimensional spaces. On the other hand, if α is finite, then all tuples are nonzero only finitely often, so the direct sum and direct product coincide; any finite dimensional vector space is isomorphic to its dual space, though usually not canonically so. In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...

Continuous dual space

When dealing with topological vector spaces, 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. The continuous dual of a vector space V is denoted V′. When the context is clear, the continuous dual may just be called the dual. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

The continuous dual V′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space. The norm ||φ|| of a continuous linear functional on V is defined by Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

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, the dual of a normed space over a complete field is necessarily complete.

For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, 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. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... 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 (see linear approximation). ...

Examples

Let 1 < p < ∞ be a real number and consider the Banach space l^p of all sequences a = (a_n) for which In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

is finite. Define the number q by 1/p + 1/q = 1. Then the continuous dual of L^p is naturally identified with L^q: given an element φ ∈ (L^p)', the corresponding element of L^q 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 a = (a_n) ∈ L^q, the corresponding continuous linear functional φ on L^p is defined by φ(b) = ∑_n a_n b_n for all b = (b_n) ∈ L^p (see Hölder's inequality). In mathematical analysis, Hölders inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ...

In a similar manner, the continuous dual of L¹ is naturally identified with L^∞. Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremums norm) and c₀ (the sequences converging to zero) are both naturally identified with L¹. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ...

Further properties

If V is a Hilbert space, then its continuous dual is a Hilbert space which is anti-isomorphic to V. This is the content of the Riesz representation theorem, and gives rise to the bra-ket notation used by physicists in the mathematical formulation of quantum mechanics. In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Fig. ...

In analogy with the case of the algebraic double dual, there is always a naturally defined injective continuous linear operator Ψ : V → V″ from V into its continuous double dual V″. This map is in fact an isometry, meaning ||Ψ(x)|| = ||x|| for all x in V. Spaces for which the map Ψ is a bijection are called reflexive. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... A bijective function. ... This page concerns the reflexivity of a Banach space. ...

The continuous dual can be used to define a new topology on V, called the weak topology. In mathematics, weak topology is an alternative term for initial topology. ...

If the dual of V is separable, then so is the space V itself. The converse is not true; the space l₁ is separable, but its dual is l_∞, which is not separable. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...