Dual space

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

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The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.

Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938. [1]

Algebraic dual space

Given any vector space over a field , the (algebraic) dual space [2] (alternatively denoted by [3] or [4] [5] ) [nb 1] is defined as the set of all linear maps (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted . [6] The dual space itself becomes a vector space over when equipped with an addition and scalar multiplication satisfying:

for all , , and .

Elements of the algebraic dual space are sometimes called covectors or one-forms .

The pairing of a functional in the dual space and an element of is sometimes denoted by a bracket: [7] or . [8] This pairing defines a nondegenerate bilinear mapping [nb 2] called the natural pairing.

Finite-dimensional case

If V is finite-dimensional, then V has the same dimension as V. Given a basis {e1, ..., en} in V, it is possible to construct a specific basis in V, called the dual basis . This dual basis is a set {e1, ..., en} of linear functionals on V, defined by the relation

for any choice of coefficients ciF. 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 is the Kronecker delta symbol. This property is referred to as bi-orthogonality property.

For example, if V is R2, let its basis be chosen as {e1 = (1/2, 1/2), e2 = (0, 1)}. The basis vectors are not orthogonal to each other. Then, e1 and e2 are one-forms (functions that map a vector to a scalar) such that e1(e1) = 1, e1(e2) = 0, e2(e1) = 0, and e2(e2) = 1. (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as

Solving this equation shows the dual basis to be {e1 = (2, 0), e2 = (−1, 1)}. Because e1 and e2 are functionals, they can be rewritten as e1(x, y) = 2x and e2(x, y) = −x + y. In general, when V is Rn, if E = (e1, ..., en) is a matrix whose columns are the basis vectors and Ê = (e1, ..., en) is a matrix whose columns are the dual basis vectors, then

where In is an identity matrix of order n. The biorthogonality property of these two basis sets allows any point xV to be represented as

even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product and the corresponding duality pairing are introduced, as described below in § Bilinear products and dual spaces .

In particular, Rn can be interpreted as the 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. This is because a functional maps every n-vector x into a real number y. Then, seeing this functional as a matrix M, and x, y as a n×1 matrix and a 1×1 matrix (trivially, a real number) respectively, if Mx = y then, by dimension reasons, M must be a 1×n matrix; that is, M must be a row vector.

If V consists of the space of geometrical vectors in the plane, then the level curves of an element of V form a family of parallel lines in V, because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. 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, it suffices to determine which of the lines the vector lies on. Informally, this "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. [9]

Infinite-dimensional case

If V is not finite-dimensional but has a basis [nb 3] eα indexed by an infinite set A, then the same construction as in the finite-dimensional case yields linearly independent elements eα (αA) of the dual space, but they will not form a basis.

For instance, the space R, whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers N: for iN, ei is the sequence consisting of all zeroes except in the i-th position, which is 1. The dual space of R is (isomorphic to) RN, the space of all sequences of real numbers: each real sequence (an) defines a function where the element (xn) of R is sent to the number

which is a finite sum because there are only finitely many nonzero xn. The dimension of R is countably infinite, whereas RN does not have a countable basis.

This observation generalizes to any [nb 3] infinite-dimensional vector space V over any field F: a choice of basis {eα : αA} identifies V with the space (FA)0 of functions f : A → F such that fα = f(α) is nonzero for only finitely many αA, where such a function f is identified with the vector

in V (the sum is finite by the assumption on f, and any vV may be written in this way by the definition of the basis).

The dual space of V may then be identified with the space FA of all functions from A to F: a linear functional T on V is uniquely determined by the values θα = T(eα) it takes on the basis of V, and any function θ : AF (with θ(α) = θα) defines a linear functional T on V by

Again the sum is finite because fα is nonzero for only finitely many α.

The set (FA)0 may be identified (essentially by definition) with the direct sum 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, FA is (again by definition), the direct product 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.

Considering cardinal numbers, denoted here as absolute values, one has thus for a F-vector space V that has an infinite basis A

It follows that, if a vector space is not finite-dimensional, then the axiom of choice implies that the algebraic dual space is always of larger dimension (as a cardinal number) than the original vector space (since, if two bases have the same cardinality, the spanned vector spaces have the same cardinality). This is in 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. [10] 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 wV to v, w. In other words, the bilinear form determines a linear mapping

defined by

If the bilinear form is 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 from V to a subspace of V (resp., all of V if V is finite dimensional) defines a unique nondegenerate bilinear form on V by

Thus there is a one-to-one correspondence between isomorphisms of V to a subspace 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 forms instead of bilinear forms. In that case, a given sesquilinear form ·,· determines an isomorphism of V with the complex conjugate of the dual space

The conjugate space V can be identified with the set of all additive complex-valued functionals f : VC such that

Injection into the double-dual

There is a natural homomorphism from into the double dual , defined by for all . In other words, if is the evaluation map defined by , then is defined as the map . This map is always injective; [nb 3] it is an isomorphism if and only if is finite-dimensional. [11] Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous double duals, not to their algebraic double duals.

Transpose of a linear map

If f : VW is a linear map, then the transpose (or dual) f : WV is defined by

for every . The resulting functional in is called the pullback of along .

The following identity holds for all and :

where the bracket [·,·] on the left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose, [12] and is formally similar to the definition of the adjoint.

The assignment ff 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) = gf. In the language of category theory, 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. It is possible to identify (f) with f using the natural injection into the double dual.

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 the transpose matrix AT with respect to the dual bases of W and V, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f 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 of S in V, denoted here S0, is the collection of linear functionals fV such that [f, s] = 0 for all sS. That is, S0 consists of all linear functionals f : VF such that the restriction to S vanishes: f|S = 0. Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.

The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: , and the annihilator of the whole space is just the zero covector: . Furthermore, the assignment of an annihilator to a subset of V reverses inclusions, so that if STV, then

If A and B are two subsets of V then

and equality holds provided V is finite-dimensional. If Ai 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 then

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 VV∗∗. In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

If W is a subspace of V then the quotient space V/W is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional f : VF factors through V/W if and only if W is in the kernel of f. There is thus an isomorphism

As a particular consequence, if V is a direct sum of two subspaces A and B, then V is a direct sum of A0 and B0.

Continuous dual space

When dealing with topological vector spaces, the continuous linear functionals from the space into the base field (or ) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space , denoted by . 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 maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".

For a topological vector space its continuous dual space, [13] or topological dual space, [14] or just dual space [13] [14] [15] [16] (in the sense of the theory of topological vector spaces) is defined as the space of all continuous linear functionals .

Properties

If X is a Hausdorff topological vector space (TVS), then the continuous dual space of X is identical to the continuous dual space of the completion of X. [1]

Topologies on the dual

There is a standard construction for introducing a topology on the continuous dual of a topological vector space . Fix a collection of bounded subsets of . This gives the topology on of uniform convergence on sets from or what is the same thing, the topology generated by seminorms of the form

where is a continuous linear functional on , and runs over the class

This means that a net of functionals tends to a functional in if and only if

Usually (but not necessarily) the class is supposed to satisfy the following conditions:

If these requirements are fulfilled then the corresponding topology on is Hausdorff and the sets

form its local base.

Here are the three most important special cases.

If is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on is normed (in fact a Banach space if the field of scalars is complete), with the norm

Each of these three choices of topology on leads to a variant of reflexivity property for topological vector spaces:

Examples

Let 1 < p < ∞ be a real number and consider the Banach space p of all sequences a = (an) for which

Define the number q by 1/p + 1/q = 1. Then the continuous dual of p is naturally identified with q: given an element , the corresponding element of q is the sequence where denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a = (an) ∈ q, the corresponding continuous linear functional on p is defined by

for all b = (bn) ∈ p (see Hölder's inequality).

In a similar manner, the continuous dual of 1 is naturally identified with (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremum norm) and c0 (the sequences converging to zero) are both naturally identified with 1.

By the Riesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which is anti-isomorphic to the original space. This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.

By the Riesz–Markov–Kakutani representation theorem, the continuous dual of certain spaces of continuous functions can be described using measures.

Transpose of a continuous linear map

If T : V → W is a continuous linear map between two topological vector spaces, then the (continuous) transpose T′ : W′ → V′ is defined by the same formula as before:

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

When V and W are normed spaces, the norm of the transpose inL(W′, V′) is equal to that of T in L(V, W). Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map T has dense range if and only if the transpose T′ is injective.

When T is a compact linear map between two Banach spaces V and W, then the transpose T′ is compact. This can be proved using the Arzelà–Ascoli theorem.

When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′. For every bounded linear map T on V, the transpose and the adjoint operators are linked by

When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with"compatible" topologies: for example, when for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ topology σ(X′, X) of pointwise convergence on X. The transpose T′ is continuous from β(W′, W) to β(V′, V), or from σ(W′, W) to σ(V′, V).

Annihilators

Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in V′,

Then, the dual of the quotient V/W can be identified with W, and the dual of W can be identified with the quotient V′/W. [20] Indeed, let P denote the canonical surjection from V onto the quotient V/W; then, the transpose P′ is an isometric isomorphism from (V/W)′ into V′, with range equal to W. If j denotes the injection map from W into V, then the kernel of the transpose j′ is the annihilator of W:

and it follows from the Hahn–Banach theorem that j′ induces an isometric isomorphism V′/WW′.

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 1 is separable, but its dual is not.

Double dual

This is a natural transformation of vector addition from a vector space to its double dual. <x1, x2>  denotes the ordered pair of two vectors. The addition + sends x1 and x2 to x1 + x2. The addition +' induced by the transformation can be defined as
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This is a natural transformation of vector addition from a vector space to its double dual. x1, x2 denotes the ordered pair of two vectors. The addition + sends x1 and x2 to x1 + x2. The addition +′ induced by the transformation can be defined as for any in the dual space.

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : VV′′ from a normed space V into its continuous double dual V′′, defined by

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ‖ Ψ(x) ‖ = ‖ x for all xV. Normed spaces for which the map Ψ is a bijection are called reflexive.

When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every xV, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual V′ of the continuous dual, again as a consequence of the Hahn–Banach theorem. [nb 4]

Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual V′, so that the continuous double dual V′′ is not uniquely defined as a set. Saying that Ψ maps from V to V′′, or in other words, that Ψ(x) is continuous on V′ for every xV, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings

be continuous for the chosen topology on V′. Further, there is still a choice of a topology on V′′, 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

Notes

  1. For used in this way, see An Introduction to Manifolds (Tu 2011 , p. 19). This notation is sometimes used when is reserved for some other meaning. For instance, in the above text, is frequently used to denote the codifferential of , so that represents the pullback of the form . Halmos (1974 , p. 20) uses to denote the algebraic dual of . However, other authors use for the continuous dual, while reserving for the algebraic dual (Trèves 2006 , p. 35).
  2. In many areas, such as quantum mechanics, ·,· is reserved for a sesquilinear form defined on V × V.
  3. 1 2 3 Several assertions in this article require the axiom of choice for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that RN has a basis. It is also needed to show that the dual of an infinite-dimensional vector space V is nonzero, and hence that the natural map from V to its double dual is injective.
  4. If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}.

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