Dual space

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.

Table of contents

1 Algebraic Dual Space 1.1 Examples 1.2 Transpose of a linear map 1.3 Bilinear products and dual spaces 1.4 Injection into the double-dual 2 Continuous Dual Space 2.5 Examples 2.6 Further properties

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., 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:

(φ + ψ)(x) = φ(x) + ψ(x)

(aφ) (x) = a φ(x)

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.

Examples

If the dimension of V is finite, then V* has the same dimension as V; if {e₁,...,e_n} is a basis for V, then the associated dual basis {e¹,...,eⁿ} of V* is given by

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 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 ∑_na_nx_n.

Transpose of a linear map

If f: V -> W is a linear map, we may define its transpose ^tf : W* -> V* by

^tf (φ) = φ o f    for every φ in W*.

The assignment f |-> ^tf 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 finite-dimensional. If the linear map f is represented by the matrix A with respect to two bases of V and W, then ^tf is represented by the transposed matrix ^tA with respect to the dual bases of W* and V*. If g: W -> X is another linear map, we have ^t(g o f) = ^tf o ^tg. 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 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

= (Φ(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 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.

Continuous Dual Space

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

||φ|| = sup { |φ(x)| : ||x|| ≤ 1 }

This turns the continuous dual into a normed vector space, indeed into a Banach space.

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.

Examples

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 = (a_n) for which

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 c₀ (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.