DUAL SPACE

(Redirected from Dual vector space)
In mathematics, any vector space ''V'' has a corresponding 'dual vector space' (or just dual space for short) consisting of all linear functionals on ''V''. 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, the dual space is an important concept in the study of functional analysis.
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 there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a ''continuous dual space''.

Contents

Algebraic dual space

The finite dimensional case

The infinite dimensional case

Bilinear products and dual spaces

Injection into the double-dual

Transpose of a linear map

Continuous dual space

Examples

Further properties

Notes

See also

Algebraic dual space

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 maps 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:
:$(phi\; +\; psi\; )(\; x\; )\; =\; phi\; (\; x\; )\; +\; psi\; (\; x\; )\; ,$
:$(\; a\; phi\; )\; (\; x\; )\; =\; a\; phi\; (\; x\; )\; ,$
for all $phi,\; psi$ in ''V''
★ , ''a'' in ''F'' and ''x'' in ''V''.
In the language of tensors, elements of ''V'' are sometimes called contravariant vectors, and elements of ''V''
★ , covariant vectors, 'covectors' or 'one-forms'.

The finite dimensional case

If ''V'' is finite-dimensional, then ''V''
★ has the same dimension as ''V'';
if {'e'₁,...,'e'_''n''} is a basis for ''V'', then the associated ''dual basis'' {'e'¹,...,'e'^''n''} of ''V''
★ is given by
:$$
mathbf{e}^i (mathbf{e}_j)= left{egin{matrix} 1, & mbox{if }i = j \ 0, & mbox{if } i
e j end{matrix}
ight.

In the case of 'R'², its basis is ''B''={'e'₁=(1,0),'e'₂=(0,1)}. Then, 'e'¹, and 'e'² are one-forms (functions which map a vector to a scalar) such that 'e'¹('e'₁)=1, 'e'¹('e'₂)=0, 'e'²('e'₁)=0, and 'e'²('e'₂)=1. (Note: The superscript here is an index, not an exponent.)
Concretely, if we interpret 'R'^''n'' 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 'R'^''n'' as a linear functional by ordinary matrix multiplication.
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.

The infinite dimensional case

If ''V'' is not finite-dimensional but has a basis^[1] 'e'_''α'' indexed by an infinite set ''A'', then the same construction as in the finite dimensional case yields linearly independent elements 'e'^''α'' (''α''∈''A'') of the dual space, but they will not form a basis.
Consider, for instance, the space 'R'^∞, whose elements are those sequences of real numbers which have only finitely many non-zero entries, which has a basis indexed by the natural numbers 'N': for ''i''∈'N', 'e'_''i'' is the sequence which is zero apart from the ''i''th 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 ∑_''n''''a''_''n''''x''_''n'', which is a finite sum because there are only finitely many nonzero ''x''_''n''. The 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 {'e'_''α'':''α''∈''A''} identifies ''V'' with the space ('F'^''A'')₀ of functions ''f'':''A''→'F' such that ''f''_''α''=''f''(''α'') is nonzero for only finitely many ''α''∈''A'', where such a function ''f'' is identified with the vector
:
in ''V'' (the sum is finite by the assumption on ''f'' and any ''v''∈''V'' may be written in this way by the definition of a 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 ''θ''_''α''=''T''('e'_''α'') it takes on the basis of ''V'', and any function ''θ'':''A''→'F' (with ''θ''(''α'')=''θ''_''α'') defines linear functional ''T'' on ''V'' by
:
Again the sum is finite because ''f''_''α'' is nonzero for only finitely many ''α''.
Note that ('F'^''A'')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 'F'^''A'' is (again by definition), the direct product of infinitely many copies of 'F' indexed by ''A'', and so the identification
:$V^$
★ cong iggl(igoplus_{lphain A}mathbb{F}iggr)^
★ cong prod_{lphain A}mathbb{F}^
★ congprod_{lphain A}mathbb{F}cong mathbb{F}^A
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 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

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
:$langle\; v,w$
angle = (Phi (v))(w) ,
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 $Psi$ from ''V'' into the double dual ''V''
★
★ , defined by $(Psi(v))(phi)\; =\; phi(v)$ for all ''v'' in ''V'', $phi$ in ''V''
★ . This map $Psi$ is always injective; it is an isomorphism 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 $f:\; V\; o\; W$ is a linear map, we may define its ''transpose'' (or ''dual'') ''f''
★ : ''W''
★ $o$ ''V''
★ by
:$f^$
★ (arphi) = arphi circ f ,
where $phi$ is an element of ''W''
★ . In that case, $f^$
★ (phi) is also known as the ''pullback'' of $phi$ by ''f''.
The assignment $f\; mapsto\; ,\; f^$
★ 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 transpose 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.
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 ^t''A'' 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 by the right on row vectors. These points of view are related by the canonical inner product on 'R'^''n'', which identifies the space of column vectors with the dual space of row vectors.

Continuous dual space

''See main article 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 ''V''
★ , denoted ''V'' ′. 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. In topological contexts sometimes ''V''
★ may also be used for just the continuous dual space and the continuous dual may just be called the ''dual''.
The continuous dual ''V'' ′ of a normed vector space ''V'' (e.g., a Banach space or a Hilbert space) forms a normed vector space. A norm ||φ|| of a continuous linear functional on ''V'' is defined by
:$|phi\; |\; =\; sup\; \{\; |phi\; (\; x\; )|\; :\; |x|\; le\; 1\; \}$
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.

Examples

Let 1 < ''p'' < ∞ be a real number and consider the Banach space ''l''^''p'' of all sequences 'a' = (''a''_''n'') for which
:$|mathbf\{a\}|\_p\; =\; left\; (\; sum\_\{n=0\}^infty\; |a\_n|^p$
ight) ^{1/p}
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 a similar manner, the continuous dual of ''l''¹ is naturally identified with ''l''^∞ (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces ''c'' (consisting of all convergent sequences, with the supremums norm) and ''c''₀ (the sequences converging to zero) are both naturally identified with ''l''¹.

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 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.
The continuous dual can be used to define a new topology on ''V'', called the weak topology.
If the dual of ''V'' is separable, then so is the space ''V'' itself. The converse is not true; the space ''l''¹ is
separable, but its dual is ''l''^∞, which is not separable.

Notes

1. 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 nonempty, and hence that the natural map from ''V'' to its double dual is injective.

See also

★ Duality (mathematics)

★ Duality (projective geometry)

★ Reciprocal lattice - dual space basis, in crystallography