In mathematics, the existence of a 'dual' vector space reflects in an abstract way the relationship between row vectors (1xn) and column vectors (nx1). The construction can also take place for infinite-dimensional 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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1.1 Examples
2 Continuous Dual Space1.2 Transpose of a linear map 1.3 Bilinear products and dual spaces 1.4 Injection into the double-dual |
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., 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:
Algebraic Dual Space
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 or one-forms.
If the dimension of V is finite,
then V* has the same dimension as V;
if {e1,...,en} is a basis for V, then the associated dual basis {e1,...,en} of V* is given by
Examples
Concretely, if we intepret 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.
If V consists of the space of geometrical vectorss (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 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. The dual of this space is Rω, the space of all sequences of real numbers. Such a sequence (an) is applied to an element (xn) of R(ω) to give the number ∑nanxn.
Transpose of a linear map
If f: V -> W is a linear map, we may define its transpose tf : W* -> V* by
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 product on V by
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.
When dealing with a normed vector space V (e.g., a Banach space or a Hilbert space), one typically is only interested in the continuous linear functionals from the space into the base field. These form a normed vector space, called the continuous dual of V, sometimes just called the dual of V. It is denoted by V '. The norm ||φ|| of a continuous linear functional on V is defined by
One may also talk about the continuous dual of an arbitrary topological vector space. This is however much harder to deal with since it will in general not be a normed vector space in any natural way.
For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, 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 space of sequences a = (an) for which
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 c0 (the sequences converging to zero) are both naturally identified with l 1.
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.Bilinear products and dual spaces
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
Continuous Dual Space
This turns the continuous dual into a normed vector space, indeed into a Banach space. Examples
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(en)) where en denotes the sequence whose nth term is 1 and all others are zero. Conversely, given an element a = (an) ∈ l q, the corresponding continuous linear functional φ on l p is defined by φ(a) = ∑n an bn for all a = (an) ∈ l p (see Hölder's inequality).Further properties