Dual space on arvau

Wikipedia / Dual space

In mathematics, every 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.

Algebraic dual space

Given any vector space over a field, the (algebraic) dual space) is defined as the set of all linear maps (linear functionals). Since linear maps are vector space homomorphisms. itself becomes a vector space over , are real numbers), the function

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

The pairing of a functional and an element of is sometimes denoted by a bracket: or . This pairing defines a nondegenerate bilinear mapping called the natural pairing.

Dual set

on that space, one can define a linearly independent called the dual set.

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

Finite-dimensional case

is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property.

be defined as the following:

because:

are one-forms.

is the identity matrix to be represented as

and the corresponding duality pairing are introduced, as described below in .

real numbers, its dual space is typically written as the space of rows as a linear functional by ordinary matrix multiplication must be a row vector.

consists of the space of geometrical vectorss in the plane, then the level curves is a vector space of any dimension, then the level sets, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.

Infinite-dimensional case

is not finite-dimensional but has a basis, then the same construction as in the finite-dimensional case yields linearly independent) of the dual space, but they will not form a basis.

, whose elements are those sequences, the space of all is sent to the number

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

This observation generalizes to any is identified with the vector

may be written uniquely in this way by the definition of the basis).

of all by

may be identified (essentially by definition) with the direct sum

is (again by definition), the direct product is a special case of a general result relating direct sums (of moduless) 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.

-vector space, the arithmetical properties of cardinal numbers where cardinalities are denoted as absolute values which can be done with an argument similar to Cantor's diagonal argument. The exact dimension of the dual is given by the Erdős–Kaplansky theorem.

Bilinear products and dual spaces

. But there is in general no natural isomorphism between these two spaces. Any bilinear form into its dual space via

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

defined by

If the bilinear form is nondegenerate by

is over the complex field, then sometimes it is more natural to consider sesquilinear forms with the complex conjugate of the dual space

Injection into the double-dual

There is a natural homomorphism is always injective; and it is always an isomorphism is finite-dimensional. 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

is a linear map, then the transpose (or dual for every . The resulting functional in is called the pullback of along .

The following identity holds for all and with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint.

produces an injective; this homomorphism is an isomorphism then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor using the natural injection into the double dual.

is represented by the matrix is represented by the transpose, which identifies the space of column vectors with the dual space of row vectors.

Quotient spaces and annihilators

. The annihilator. Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.

and

is a vector subspace. In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

then the quotient space is a vector space in its own right, and so has a dual. By the first isomorphism theorem is in the kernel is a direct sum.

Dimensional analysis

to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction. This is formalized by tensor contraction.

This arises in physics via dimensional analysis, where the dual space has inverse units. Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected. For example, in (continuous) Fourier analysis, or more broadly time–frequency analysis: given a one-dimensional vector space with a unit of time, the dual space has units of frequency: occurrences per). For example, if time is measured in seconds, the corresponding dual unit is the inverse second. Similarly, if the primal space measures length, the dual space measures inverse length. == Continuous dual space ==

When dealing with topological vector spaces, the continuous. 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".

For a topological vector space its continuous dual space, or topological dual space, or just dual space.

Important examples for continuous dual spaces are the space of compactly supported test functions the space of arbitrary distributions the Schwartz space the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.

Properties

is a Hausdorff topological vector space is identical to the continuous dual space of the completion.

Topologies on the dual

of bounded subsets of uniform convergence or what is the same thing, the topology generated by seminorms

is satisfies the following conditions:

.
.
is closed under the operation of multiplication by scalars.

form its local base.

Here are the three most important special cases.

The strong topology is the topology of uniform convergence on bounded subsets).

is a normed vector space (for example, a Banach space or a Hilbert space

The stereotype topology is the topology of uniform convergence on totally bounded sets).
The weak topology).

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

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.
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.
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

be a real number and consider the Banach space ℓ p of all sequences where , the corresponding continuous linear functional (see Hölder's inequality).

(consisting of all convergent sequences, with the supremum norm.

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

. Several properties of transposition depend upon the Hahn–Banach theorem has dense range if and only if is injective.

is a compact is compact. This can be proved using the Arzelà–Ascoli theorem.

, the transpose and the adjoint have the strong topology.

Annihilators

. denote the canonical surjection and it follows from the Hahn–Banach theorem.

Further properties

is separable is not.

Double dual

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry is a bijection are called reflexive.

is a topological vector space is not locally convex is Hausdorff of the continuous dual, again as a consequence of the Hahn–Banach theorem.

depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.