Dual space

In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). The construction can also take place for infinitedimensional spaces and gives rise to important ways of looking at measures, distributions and Hilbert space. The use of the dual space in some fashion is thus characteristic of functional analysis. It is also inherent in the Fourier transform.
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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 F, i.e., scalarvalued linear transformations on V (in this context, a "scalar" is a member of the basefield F). V* itself becomes a vector space over F under the following definition of addition and scalar multiplication:
 <math> (\phi + \psi )( x ) = \phi ( x ) + \psi ( x ) \,<math>
 <math> ( a \phi ) ( x ) = a \phi ( x ) \,<math>
for all φ, ψ 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 oneforms.
Examples
If the dimension of V is finite, 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
 <math>
e^i (e_j)= \left\{\begin{matrix} 1, & \mbox{if }i = j \\ 0, & \mbox{if } i \ne j \end{matrix}\right. <math>
Concretely, if we interpret R^{n} 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 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.
If V is infinitedimensional, then the above construction of e^{i} 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 nonzero entries. The dual of this space is R^{ω}, the space of all sequences of real numbers. Such a sequence (a_{n}) is applied to an element (x_{n}) of R^{(ω)} to give the number ∑_{n}a_{n}x_{n}.
Transpose of a linear map
If f: V > W is a linear map, we may define its transpose ^{t}f : W* → V* by
 <math>
The assignment f > ^{t}f produces an injective homomorphism 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 iff W is finitedimensional. If the linear map f is represented by the matrix A with respect to two bases of V and W, then ^{t}f is represented by the transposed matrix ^{t}A with respect to the dual bases of W* and V*. If g: W → X is another linear map, we have ^{t}(g o f) = ^{t}f o ^{t}g. 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.
Bilinear products and dual spaces
As we saw above, if V is finitedimensional, 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 nondegenerate bilinear form on V by
 <math> \langle v,w \rangle = (\Phi (v))(w) \,<math>
and conversely every such nondegenerate bilinear product on a finitedimensional space gives rise to an isomorphism from V to V*.
Injection into the doubledual
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 finitedimensional.
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.
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
 <math>\\phi \ = \sup \{ \phi ( x ) : \x\ \le 1 \}<math>
This turns the continuous dual into a normed vector space, indeed into a Banach space.
Examples
For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide.
Let 1 < p < ∞ be a real number and consider the Banach space L^{p} of all sequences a = (a_{n}) for which
 <math>\\mathbf{a}\_p = \left ( \sum_{n=0}^\infty a_n^p \right) ^{1/p}<math>
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 nth 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(a) = ∑_{n} a_{n} b_{n} for all a = (a_{n}) ∈ L^{p} (see Hölder's inequality).
In a similar manner, the continuous dual of L^{1} is naturally identified with L^{∞}. 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 antiisomorphic to V. This is the content of the Riesz representation theorem, and gives rise to the braket 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.de:Dualraum fr:Espace dual ja:双対ベクトル空間 zh:对偶空间