Dual space

The ( algebraic ) dual space is a term from the mathematical subfield of linear algebra. The dual space of a vector space over a field is the vector space of all linear maps from to. These linear maps are sometimes called covectors.

If the vector space is finite-dimensional, so it has the same dimension as its dual space. The two vector spaces are thus isomorphic.

In the functional analysis, we consider the topological dual space of a (generally infinite-dimensional ) topological vector space. This consists of all continuous linear functionals. The dual space of a dual space is called Bidualraum.

• 2.1 Topological dual space of a normed space
• 2.2 The strong dual space of a locally convex space
• 2.3 Bidualraum
• 2.4 Examples

The algebraic dual space

Definition and concept formation

Refers to a vector space over a field belonging to the dual space, ie the set of all linear maps from to. Its elements are called depending on the context, functional, linear forms or 1-forms. In particular, in physics one likes to use the language of tensor algebra; then the names of the elements of contravariant that. covariant vectors or covectors of The figure is a nondegenerate bilinear form and is called dual pairing.

Dual space is a vector space

The subsequent definition of addition and scalar multiplication of on is itself a vector space over the field.

For this purpose, the vectorial addition

And the scalar multiplication

Defined.

Basis of the dual space

Is one - dimensional vector space so also -dimensional. It is therefore necessary.

Let be a basis of, then that means with

The dual base to the base and is a base of the dual compartment. With the help of the dual pairing, the effect of dual basis vectors to basis vectors Write clearly with the Kronecker delta:

By identifying each linear form of the algebraic dual space with its core, so the solution set of homogeneous linear equation, you get into the Projective Geometry to a duality between points and hyperplanes of the projective space. This duality is represented in the article " projective coordinate system."

If, however, an infinite-dimensional vector space, so can not construct dual basis in this way in general. Thus, if a base of infinite-dimensional vector space. Then one can consider the linear mapping. This is an element of the dual space, but it can not be considered a finite linear combination of the pose. Therefore no system of generators of form.

Dual imaging

Is a linear map between vector spaces and, then by

A linear map between the dual spaces and given. It is called the dual to Figure.

Are - linear mappings, the following applies

And for all

By assigning a so - linear mapping is given.

If an injective linear map, then the dual map is surjective. If, however, surjective, then is injective.

Is another vector space and are linearly and then applies

Bidualraum

The dual space of the dual space of a vector space is called Bidualraum and designated. The elements of are so linear mappings that map the functional scalars from. For each of the mapping that assigns each scalar, such a mapping, that is, it is.

The figure

Is linear and injective. Therefore can always be identified by a sub-space of. This is called the natural or canonical embedding of the space in his Bidualraum.

Is finite, the following applies. In this case, even bijective and is called a canonical isomorphism between and.

The topological dual space

If the underlying vector space is a topological vector space, one can in addition to algebraic also consider the topological dual space. This is the set of continuous linear functional and is designated generally with. The distinction between algebraic and topological dual space is only important when an infinite-dimensional space, since all linear operators that are defined on a finite-dimensional topological vector space, are also continuous. Thus, the algebraic and the topological dual space are identical. If in the context of topological vector spaces from a dual space is mentioned, the topological dual space is usually meant. The study of these dual spaces is one of the main areas of functional analysis.

Topological dual space of a normed space

The considered in functional analysis spaces often carry a topology that is induced by a norm. In this case, the topological dual space is a normed vector space with the operator norm.

Since the underlying body of a normed space, either the field of real or complex numbers and thus is complete, the dual space is also complete, ie it is a Banach space, regardless of whether itself is complete.

Particularly simple is the ( topological ) dual space, if a Hilbert space. After a block, the M. Fréchet in 1907 has proved for separable and F. Riesz in 1934 for general Hilbert spaces, a real Hilbert space and its dual space are isometrically isomorphic to each other, see Theorem of Fréchet - Riesz. The commutativity of space and dual space is particularly evident in the Bra- Ket notation of Dirac expressed. This is used especially in quantum mechanics because the quantum -mechanical states are modeled by vectors in a Hilbert space.

Since every finite dimensional vector space over the real or complex numbers is isomorphic to a Hilbert space, finite dimensional spaces are always dual to itself.

The strong dual space of a locally convex space

If E is a locally convex space, so called E ' as in the case of normed spaces the space of continuous linear functionals. The award of an appropriate topology on the dual space is more complex. The following definition is applied, which results in the special case of the normalized space, the norm topology described above on the dual space:

Is limited, so defines a semi-norm on E '. The set of semi-norms, where B runs through the limited amounts of E, defines the so-called strong topology on E '. It's called E ' with the strong topology to the strong dual space and sometimes referred to him with precise, where subscript b (bounded german, French borné ) for limited stands.

The weak -* topology is also a frequently studied topology on E ', but this falls in the case of infinite-dimensional normed spaces not with the norm topology described above on the dual space together. In the theory of locally convex spaces is therefore meant by dual space is usually the strong dual space.

Bidualraum

Since the dual space of a normed space is a Banach space by the above, one can consider the dual space of the dual space, the so-called Bidualraum. Here it is interesting that there is a canonical embedding in, which is given by. (That is, each element of the original space is naturally also an element of Bidualraums ). If every element of Bidualraums can be represented by an element of, or more precisely if the canonical embedding is an isomorphism, then the Banach space is called reflexive. Reflexive spaces are easier to handle than non- reflexive, they are in some ways the most similar to Hilbert spaces. In the non- reflexive case, the canonical embedding is not surjective but still more isometric, and you typically writes. Accordingly, each normed space is contained in a Banach space; the transition from the topological degree in is a way to make the completion of a normed space.

An example of a non- reflexive space is the sequence space of all null sequences with the maximum norm. The Bidualraum can be identified in a natural way with the sequence space of bounded sequences with the supremum norm. There are non- reflexive Banach spaces in which the canonical embedding is therefore not an isomorphism, but there is another isomorphism between space and Bidualraum. An example of this is the so-called James - space, according to Robert C. James.

Examples

In the following list, another Banach space is given in the second column to the first column of a Banach space which is isometrically isomorphic to the dual space of V in the sense specified in the third column duality. Specifically, this means: Each element of defined by the formula of duality a continuous linear functional on. This produces an image, and that is linear and bijective isometric.