Reflexive space

Reflexivity is a concept from functional analysis and algebra. A space is reflexive if the natural embedding in its Bidualraum is an isomorphism, as will be explained below. Thus, a reflexive space can be identified with the dual space of its dual space.

Reflexive spaces

In the functional analysis, reflexivity is a property of normed vector spaces.

Definition

It should be a normed space ( or over ). One can show that its ( topological ) dual space is a Banach space. Its dual space is denoted by and is called Bidualraum of.

By mapping rule

Defines a continuous linear isometry, the canonical embedding. The defining equation thus reads as follows in Bilinearformschreibweise:

As isometry is injective. In addition, if is surjective, for a total of an isometric isomorphism between and, it is called a reflexive space.

Examples

  • Every finite- dimensional Banach space is reflexive.
  • According to the Riesz representation theorem every Hilbert space is reflexive.
  • Closed subspaces of reflexive spaces are reflexive.
  • For any and all of the Lebesgue spaces and all Sobolev spaces are reflexive for all open subsets.
  • For all the sequence spaces are reflexive.
  • The Banach spaces are not reflexive.
  • Be a null sequence set, and This Banach space was designed by Robert C. James in 1951 and is an example of a non-reflexive Banach space, but which is isometrically isomorphic to its Bidualraum.

Reflexivitätskriterien

A Banach space is reflexive,

  • ( Set of Eberlein - Šmulian ) if every bounded sequence has a weakly convergent subsequence.
  • ( Set of James ) if every continuous linear functional assumes its norm on the unit sphere.

Properties of reflexive spaces

Each reflexive normed space is a Banach space, because it is by definition isomorphic to the full Bidualraum. In reflexive Banach spaces the closed unit ball ( generally, any bounded and weakly closed subset ) is weakly compact, ie compact with respect to the weak topology (this follows directly from the set of Banach Alaoglu about the weak * - compactness of the unit ball of a reflexive Banach space Bidualraum ).

This property characterizes the reflexive spaces: A Banach space is reflexive if its unit ball is weakly compact.

In particular, each limited power in a reflexive space has a weakly convergent subnet. With the set of Eberlein - Šmulian follows that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. Next Permanenzaussagen following apply:

  • X is reflexive if X is reflexive and complete.
  • If X is reflexive and a closed subspace, then and reflexive.

Applications

Often together with the Sobolev embedding sets provides the existence of weakly convergent subsequences limited impact solutions of variational problems and thus partial differential equations.

Reflexive locally convex spaces

Provides you the dual space of a locally convex space X with the strong topology, we obtain an injective, continuous, linear map. is called reflexive if a topological isomorphism and semi-reflexive if it is surjective. In contrast to the case of normed spaces is not automatically a topological isomorphism in the semi-reflexive case. There are the following levels:

  • A locally convex space is reflexive if and only if every weakly closed bounded set is weakly compact.
  • A locally convex space is reflexive if it is reflexive and quasi-barreled.

Reflexive modules

If a module over a commutative ring with identity, then the module of the dual module is called; the module is called Bidualmodul. There is a canonical map

Which in general is neither injective nor surjective. If it is an isomorphism, so is called reflexive.

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