Dual space explained

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.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the .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.

Algebraic dual space

over a field , the (algebraic) dual space ^[1] (alternatively denoted by ^[2] or ^{[3] [4]})^[5] is defined as the set of all linear maps

\varphi:V\toF

(linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted .^[2] The dual space itself becomes a vector space over

F

when equipped with an addition and scalar multiplication satisfying:for all ,

x\inV

, and . For example, if we express the vector space as the set of vectors (where and are real numbers), the function

$$f(r\backslash mathbf\; +\; s\backslash mathbf)\; =\; r\; +\; s$$

is an element of

, since it is -linear and maps vectors in to elements of .

Elements of the algebraic dual space

are sometimes called covectors, one-forms, or linear forms.

The pairing of a functional

in the dual space

V^*

and an element of is sometimes denoted by a bracket: ^[6] or . This pairing defines a nondegenerate bilinear mapping^[7]

\langle ⋅ , ⋅ \rangle:V x V^*\toF

called the natural pairing.

Dual set

See main article: article and Dual basis. Given a vector space

and a basis on that space, one can define a linearly independent set in called the dual set. Each vector in corresponds to a unique vector in the dual set. This correspondence yields an injection .

If

is finite-dimensional, the dual set is a basis, called the dual basis, and the injection is an isomorphism.

Finite-dimensional case

If

is finite-dimensional and has a basis, in , the dual basis is a set of linear functionals on , defined by the relation$$\backslash mathbf^i(c^1\; \backslash mathbf\_1+\backslash cdots+c^n\backslash mathbf\_n)\; =\; c^i,\; \backslash quad\; i=1,\backslash ldots,n$$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$$\backslash mathbf^i(\backslash mathbf\_j)\; =\; \backslash delta^\_$$where is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property.

Consider

the basis of V. Let be defined as the following: .

These are a basis of

because:

1. The

are linear functionals, which map such as and to scalars and . Then also, and . Therefore, for .

1. Suppose

. Applying this functional on the basis vectors of successively, lead us to (The functional applied in results in ). Therefore, is linearly independent on .

1. Lastly, consider

. Then $$g(x)=g(\backslash alpha\_1\backslash mathbf\_1\; +\; \backslash dots\; +\; \backslash alpha\_n\backslash mathbf\_n)=\backslash alpha\_1g(\backslash mathbf\_1)\; +\; \backslash dots\; +\; \backslash alpha\_ng(\backslash mathbf\_n)=\backslash mathbf^1(x)g(\backslash mathbf\_1)\; +\; \backslash dots\; +\; \backslash mathbf^n(x)g(\backslash mathbf\_n)$$ so . So generates .Hence, it is a basis of .

For example, if

is , let its basis be chosen as . The basis vectors are not orthogonal to each other. Then, and are one-forms (functions that map a vector to a scalar) such that , , , and . (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation asSolving for the unknown values in the first matrix shows the dual basis to be . Because and are functionals, they can be rewritten as and .

In general, when

is , if is a matrix whose columns are the basis vectors and is a matrix whose columns are the dual basis vectors, then$$\backslash hat^\backslash textrm\backslash cdot\; E\; =\; I\_n,$$where is the identity matrix of order . The biorthogonality property of these two basis sets allows any point 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 .

In particular,

can be interpreted as the space of columns of real numbers, its dual space is typically written as the space of rows of real numbers. Such a row acts on as a linear functional by ordinary matrix multiplication. This is because a functional maps every -vector into a real number . Then, seeing this functional as a matrix , and as an matrix, and a matrix (trivially, a real number) respectively, if then, by dimension reasons, must be a matrix; that is, must be a row vector.

If

consists of the space of geometrical vectors in the plane, then the level curves of an element of form a family of parallel lines in , 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 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 is a vector space of any dimension, then the level sets of a linear functional in are parallel hyperplanes in , and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.

Infinite-dimensional case

If

is not finite-dimensional but has a basis^[8] indexed by an infinite set , then the same construction as in the finite-dimensional case yields linearly independent elements ( ) of the dual space, but they will not form a basis.

For instance, consider the space

, 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 . For , is the sequence consisting of all zeroes except in the -th position, which is 1.The dual space of is (isomorphic to) , the space of all sequences of real numbers: each real sequence defines a function where the element of is sent to the number

$$\backslash sum\_n\; a\_nx\_n,$$

which is a finite sum because there are only finitely many nonzero

. The dimension of is countably infinite, whereas does not have a countable basis.

This observation generalizes to any^[8] infinite-dimensional vector space

over any field : a choice of basis identifies with the space of functions such that is nonzero for only finitely many , where such a function is identified with the vector

$$\backslash sum\_\; f\_\backslash alpha\backslash mathbf\_\backslash alpha$$

in

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

The dual space of

may then be identified with the space of all functions from to : a linear functional on is uniquely determined by the values it takes on the basis of , and any function (with ) defines a linear functional on by

$$T\backslash left\; (\backslash sum\_\; f\_\backslash alpha\; \backslash mathbf\_\backslash alpha\backslash right)\; =\; \backslash sum\_\; f\_\backslash alpha\; T(e\_\backslash alpha)\; =\; \backslash sum\_\; f\_\backslash alpha\; \backslash theta\_\backslash alpha.$$

Again, the sum is finite because

is nonzero for only finitely many .

The set

may be identified (essentially by definition) with the direct sum of infinitely many copies of (viewed as a 1-dimensional vector space over itself) indexed by , i.e. there are linear isomorphisms$$V\backslash cong\; (F^A)\_0\backslash cong\backslash bigoplus\_\; F.$$

On the other hand,

is (again by definition), the direct product of infinitely many copies of indexed by , and so the identification$$V^*\; \backslash cong\; \backslash left\; (\backslash bigoplus\_F\backslash right)^*\; \backslash cong\; \backslash prod\_F^*\; \backslash cong\; \backslash prod\_F\; \backslash cong\; F^A$$is a special case of a general result relating direct sums (of modules) to direct products.

If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. 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.

The proof of this inequality between dimensions results from the following.

If

is an infinite-dimensional -vector space, the arithmetical properties of cardinal numbers implies that =|V^\ast|=\mathrm(|\mathrm(V^\ast)|, |F|),where cardinalities are denoted as absolute values. For proving that it suffices to prove that which can be done with an argument similar to Cantor's diagonal argument.^[9] The exact dimension of the dual is given by the Erdős–Kaplansky theorem.

Bilinear products and dual spaces

If

is finite-dimensional, then is isomorphic to . But there is in general no natural isomorphism between these two spaces. Any bilinear form on gives a mapping of into its dual space via

$$v\backslash mapsto\; \backslash langle\; v,\; \backslash cdot\backslash rangle$$

where the right hand side is defined as the functional on

taking each to . In other words, the bilinear form determines a linear mapping

$$\backslash Phi\_\; :\; V\backslash to\; V^*$$

defined by

$$\backslash left[\backslash Phi\_\{\backslash langle\backslash cdot,\backslash cdot\backslash rangle\}(v),\; w\backslash right]\; =\; \backslash langle\; v,\; w\backslash rangle.$$

If the bilinear form is nondegenerate, then this is an isomorphism onto a subspace of

.If is finite-dimensional, then this is an isomorphism onto all of . Conversely, any isomorphism from to a subspace of (resp., all of if is finite dimensional) defines a unique nondegenerate bilinear form on by

$$\backslash langle\; v,\; w\; \backslash rangle\_\backslash Phi\; =\; (\backslash Phi\; (v))(w)\; =\; [\backslash Phi\; (v),\; w].\backslash ,$$

Thus there is a one-to-one correspondence between isomorphisms of

to a subspace of (resp., all of) and nondegenerate bilinear forms on .

If the vector space

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 with the complex conjugate of the dual space

$$\backslash Phi\_\; :\; V\backslash to\; \backslash overline.$$The conjugate of the dual space

can be identified with the set of all additive complex-valued functionals such that$$f(\backslash alpha\; v)\; =\; \backslash overlinef(v).$$

Injection into the double-dual

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;^[8] and it is always an isomorphism if is finite-dimensional.^[10] 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 isomorphic to their algebraic double duals, but instead to their continuous double duals.

Transpose of a linear map

See main article: Transpose of a linear map. If

is a linear map, then the transpose (or dual) is defined by$$f^*(\backslash varphi)\; =\; \backslash varphi\; \backslash circ\; f\; \backslash ,$$for every

\varphi\inW^*

. The resulting functional

f^*(\varphi)

in

V^*

is called the pullback of

\varphi

along

f

.

The following identity holds for all

and :$$[f^*(\backslash varphi),\backslash ,\; v]\; =\; [\backslash varphi,\backslash ,\; f(v)],$$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,^[11] and is formally similar to the definition of the adjoint.

The assignment

produces an injective linear map between the space of linear operators from to and the space of linear operators from to ; this homomorphism is an isomorphism if and only if is finite-dimensional.If 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 .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 to itself.It is possible to identify with using the natural injection into the double dual.

If the linear map

is represented by the matrix with respect to two bases of and , then is represented by the transpose matrix with respect to the dual bases of and , hence the name.Alternatively, as is represented by acting on the left on column vectors, 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 , which identifies the space of column vectors with the dual space of row vectors.

Quotient spaces and annihilators

Let

be a subset of .The annihilator of in , denoted here , is the collection of linear functionals such that for all .That is, consists of all linear functionals such that the restriction to vanishes: .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 reverses inclusions, so that if , then$$\backslash \; \backslash subseteq\; T^0\; \backslash subseteq\; S^0\; \backslash subseteq\; V^*\; .$$

If

and are two subsets of then$$A^0\; +\; B^0\; \backslash subseteq\; (A\; \backslash cap\; B)^0\; .$$If is any family of subsets of indexed by belonging to some index set , then$$\backslash left(\backslash bigcup\_\; A\_i\; \backslash right)^0\; =\; \backslash bigcap\_\; A\_i^0\; .$$In particular if and are subspaces of then$$(A\; +\; B)^0\; =\; A^0\; \backslash cap\; B^0$$and^[8] $$(A\; \backslash cap\; B)^0\; =\; A^0\; +\; B^0\; .$$

If

is finite-dimensional and is a vector subspace, then$$W^\; =\; W$$after identifying with its image in the second dual space under the double duality isomorphism . In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

If

is a subspace of then the quotient space is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional factors through if and only if is in the kernel of . There is thus an isomorphism As a particular consequence, if is a direct sum of two subspaces and , then is a direct sum of and .

Dimensional analysis

The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector

can be paired with a covector by the natural pairing to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction. Thus while the direct sum is a -dimensional space (if is -dimensional), behaves as an -dimensional space, in the sense that its dimensions can be canceled against the dimensions of . This is formalized by tensor contraction. , the dual space has units of frequency: occurrences per unit of time (units of ). For example, if time is measured in seconds, the corresponding dual unit is the inverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to . Similarly, if the primal space measures length, the dual space measures inverse length.

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". its continuous dual space, or topological dual space, or just dual space (in the sense of the theory of topological vector spaces) is defined as the space of all continuous linear functionals . and its dual the space of arbitrary distributions (generalized functions); the space of arbitrary test functions and its dual the space of compactly supported distributions; and the space of rapidly decreasing test functions the Schwartz space, and its dual the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.

Properties

If

is a Hausdorff topological vector space (TVS), then the continuous dual space of is identical to the continuous dual space of the completion of .

Topologies on the dual

See main article: Polar topology and Dual system.

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

$$\backslash |\backslash varphi\backslash |\_A\; =\; \backslash sup\_\; |\backslash varphi(x)|,$$

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

$$\backslash text\; A\backslash in\backslash mathcal\backslash qquad\; \backslash |\backslash varphi\_i-\backslash varphi\backslash |\_A\; =\; \backslash sup\_\; |\backslash varphi\_i(x)-\backslash varphi(x)|\backslash underset\; 0.$$

Usually (but not necessarily) the class

is supposed to satisfy the following conditions:

• Each point

of belongs to some set .

• Each two sets

and are contained in some set . is closed under the operation of multiplication by scalars.

If these requirements are fulfilled then the corresponding topology on

is Hausdorff and the sets

$$U\_A\; ~=~\; \backslash left\; \backslash ,\backslash qquad\; \backslash text\; A\; \backslash in\; \backslash mathcal$$

form its local base.

Here are the three most important special cases.

• The strong topology on

is the topology of uniform convergence on bounded subsets in (so here can be chosen as the class of all bounded subsets in ).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$$\backslash |\backslash varphi\backslash |\; =\; \backslash sup\_\; |\backslash varphi(x)|.$$

• The stereotype topology on

is the topology of uniform convergence on totally bounded sets in (so here can be chosen as the class of all totally bounded subsets in ).

• The weak topology on

is the topology of uniform convergence on finite subsets in (so here can be chosen as the class of all finite subsets in ).

Each of these three choices of topology on

leads to a variant of reflexivity property for topological vector spaces:

• If

is endowed with the strong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called reflexive.

• If

is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of stereotype spaces: the spaces reflexive in this sense are called stereotype.

• If

is endowed with the weak topology, then the corresponding reflexivity is presented in the theory of dual pairs: the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.

Examples

Let 1 <

< ∞ be a real number and consider the Banach space ℓ^&thinsp;p of all sequences for which

Define the number

by . Then the continuous dual of is naturally identified with : given an element , the corresponding element of is the sequence where

e_n

denotes the sequence whose -th term is 1 and all others are zero. Conversely, given an element , the corresponding continuous linear functional

\varphi

on is defined by

$$\backslash varphi\; (\backslash mathbf)\; =\; \backslash sum\_n\; a\_n\; b\_n$$

for all

(see Hölder's inequality).

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 (consisting of all convergent sequences, with the supremum norm) and (the sequences converging to zero) are both naturally identified with .

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

See also: Transpose of a 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:

$$T\text{'}(\backslash varphi)\; =\; \backslash varphi\; \backslash circ\; T,\; \backslash quad\; \backslash varphi\; \backslash in\; W\text{'}.$$

The resulting functional

is in . The assignment produces a linear map between the space of continuous linear maps from to and the space of linear maps from to .When and are composable continuous linear maps, then

$$(U\; \backslash circ\; T)\text{'}\; =\; T\text{'}\; \backslash circ\; U\text{'}.$$

When

and are normed spaces, the norm of the transpose in is equal to that of in .Several properties of transposition depend upon the Hahn–Banach theorem.For example, the bounded linear map has dense range if and only if the transpose is injective.

When

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

When

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

$$i\_V\; \backslash circ\; T^*\; =\; T\text{'}\; \backslash circ\; i\_V.$$

When

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

Annihilators

Assume that

is a closed linear subspace of a normed space , and consider the annihilator of in ,

$$W^\backslash perp\; =\; \backslash .$$

Then, the dual of the quotient

can be identified with , and the dual of can be identified with the quotient .Indeed, let denote the canonical surjection from onto the quotient . Then the transpose is an isometric isomorphism from into , with range equal to .If denotes the injection map from into , then the kernel of the transpose is the annihilator of :$$\backslash ker\; (j\text{'})\; =\; W^\backslash perp$$and it follows from the Hahn–Banach theorem that induces an isometric isomorphism .

Further properties

If the dual of a normed space

is separable, then so is the space 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 into its continuous double dual , defined by

$$\backslash Psi(x)(\backslash varphi)\; =\; \backslash varphi(x),\; \backslash quad\; x\; \backslash in\; V,\; \backslash \; \backslash varphi\; \backslash in\; V\text{'}\; .$$

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning

for all .Normed spaces for which the map is a bijection are called reflexive.

When

is a topological vector space then ( ) can still be defined by the same formula, for every , however several difficulties arise.First, when is not locally convex, the continuous dual may be equal to and the map trivial.However, if is Hausdorff and locally convex, the map is injective from to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.^[14]

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 maps from to , or in other words, that is continuous on for every , is a reasonable minimal requirement on the topology of , namely that the evaluation mappings

$$\backslash varphi\; \backslash in\; V\text{'}\; \backslash mapsto\; \backslash varphi(x),\; \backslash quad\; x\; \backslash in\; V,$$

be continuous for the chosen topology on

. Further, there is still a choice of a topology on , and continuity of depends upon this choice.As a consequence, defining reflexivity in this framework is more involved than in the normed case.

See also

• Covariance and contravariance of vectors
• Dual module
• Dual norm
• Duality (mathematics)
• Duality (projective geometry)
• Pontryagin duality
• Reciprocal lattice – dual space basis, in crystallography

Bibliography

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Notes and References

1. p. 37, §2.1.3
2. p. 19, §3.1
3. p. 101, §3.94
4. p. 20, §13
5. For

   V^\lor

   used in this way, see An Introduction to Manifolds .This notation is sometimes used when

   ( ⋅ )^*

   is reserved for some other meaning.For instance, in the above text,

   F^*

   is frequently used to denote the codifferential of

   F

   , so that

   F^*\omega

   represents the pullback of the form

   \omega

   .

   uses

   V'

   to denote the algebraic dual of

   V

   . However, other authors use

   V'

   for the continuous dual, while reserving

   V^*

   for the algebraic dual .
6. p. 21, §14
7. In many areas, such as quantum mechanics,

   \langle ⋅ , ⋅ \rangle

   is reserved for a sesquilinear form defined on

   V x V

   .
8. 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

   \R^\N

   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.
9. Book: Elements of mathematics: Algebra I, Chapters 1 - 3. Nicolas Bourbaki. 400. Hermann. 0201006391. 1974. Addison-Wesley Publishing Company . en.
10. pp. 25, 28
11. §44
12. Web site: A mathematical formalisation of dimensional analysis. Tao . Terence . Terence Tao. 2012-12-29. Similarly, one can define

    	T-1
    V

    as the dual space to

    V^T

    ....
13. To be precise, continuous Fourier analysis studies the space of functionals with domain a vector space and the space of functionals on the dual vector space.
14. If

    V

    is locally convex but not Hausdorff, the kernel of

    \Psi

    is the smallest closed subspace containing .