Reflective space

Reflexivity is a term from functional analysis and algebra . A space is reflexive when the natural embedding in its dual space is an isomorphism , as explained below. A reflective space can thus be identified with the dual space of its dual space.

Reflexive spaces

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

definition

Let it be a standardized space (above or ). One can show that his (topological) dual space is a Banach space . Its dual space is denoted by and is called the dual space of . ${\displaystyle (X,\|\cdot \|_{X})}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X'}$ ${\displaystyle \left(X'\right)'}$ ${\displaystyle X''}$ ${\displaystyle X}$

By the mapping rule

${\displaystyle X\to X'',x\mapsto [x'\mapsto x'(x)]}$

a continuous linear isometry is defined, the canonical embedding . The defining equation of reads as follows in bilinear form notation: ${\displaystyle J_{X}\colon X\to X''}$ ${\displaystyle J_{X}}$

${\displaystyle \langle J_{X}x,x'\rangle _{X'}=\langle x',x\rangle _{X}\quad \forall x'\in X'.}$

As isometry is injective . If there is also surjective , i.e. an isometric isomorphism between and , it is called a reflexive space. ${\displaystyle J_{X}}$ ${\displaystyle J_{X}}$ ${\displaystyle X}$ ${\displaystyle X''}$ ${\displaystyle X}$

Examples

• Every finite-dimensional Banach space is reflexive.
• According to the Fréchet-Riesz theorem , every Hilbert space is reflexive.
• Closed sub-spaces of reflexive spaces are reflexive.
• For everyone and everyone , the Lebesgue spaces as well as all Sobolev spaces are reflexive for all open subsets . ${\displaystyle 1<p<\infty }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle L^{p}\left(\Omega \right)}$ ${\displaystyle W^{k,p}\left(\Omega \right)}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$
• For those who are sequence spaces with reflexive. ${\displaystyle 1<p<\infty }$ ${\displaystyle \ell ^{p}(\mathbb {K} )}$ ${\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} }$
• The Banach spaces are not reflexive. ${\displaystyle \ell ^{1}(\mathbb {K} ),\ell ^{\infty }(\mathbb {K} ),L^{1}(\Omega ),L^{\infty }(\Omega ),BC^{k}(\Omega )}$
• In 1951, Robert C. James constructed the James Room named after him . This is not reflexive but isometrically isomorphic to its bidual, that is, the canonical embedding of the room in its bidual is not surjective, but there is still another isometric isomorphism of the space on its bidual.

Reflexivity criteria

A Banach space is reflexive if and only if

• (Kakutani's theorem) the unit sphere is compact in the weak topology.
• ( Eberlein – Šmulian Theorem ) every bounded sequence has a weakly convergent subsequence.
• ( James Theorem ) every continuous linear functional assumes its norm on the unit sphere.
• (Šmulian, 1939) every descending sequence of non-empty, bounded, closed, and convex sets has a non-empty intersection.

The last characterization is noteworthy because it exclusively uses the Banach space itself, i.e. in particular makes no reference to the dual space (see definition) or the dual space (use of the weak topology or James theorem).

Properties of reflective spaces

Every reflexive normalized space is a Banach space, because by definition it is isomorphic to the complete bidual space. In reflexive Banach spaces the closed unit sphere (more generally any bounded and weakly closed subset) is weakly compact, i.e. H. compact with respect to the weak topology (this follows directly from the Banach-Alaoğlu theorem about the weak * -compactness of the unit sphere of the dual space of a reflexive Banach space).

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

In particular, every constrained network in a reflexive space has a weakly convergent subnet. With the Eberlein – Šmulian theorem it follows that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence . The following statements about permanence also apply:

• ${\displaystyle X}$is reflexive if and only if is reflexive and complete. ${\displaystyle X'}$ ${\displaystyle X}$
• If reflexive and a closed subspace, then and are reflexive. ${\displaystyle X}$ ${\displaystyle Y\subset X}$ ${\displaystyle Y}$ ${\displaystyle X/Y}$

Applications

Together with Sobolev's embedding theorems , the existence of weakly convergent partial sequences of bounded sequences often provides solutions to variational problems and thus partial differential equations .

Reflexive locally convex spaces

If one provides the dual space of a locally convex space X with the strong topology , one obtains an injective, continuous, linear mapping . is called reflexive when is a topological isomorphism and semi-reflexive when is surjective. In contrast to the case of normalized spaces, a topological isomorphism is not automatically in the semi-reflective case. The following rates apply: ${\displaystyle J_{X}:X\rightarrow X'',\,J_{X}(x)(x'):=x'(x)}$ ${\displaystyle X}$ ${\displaystyle J_{X}}$ ${\displaystyle J_{X}}$ ${\displaystyle J_{X}}$

• A locally convex space is semi-reflective if and only if every weakly closed bounded set is weakly compact .
• A locally convex space is reflexive if and only if it is semi-reflexive and quasitonneliert .

Reflexive modules

If a module is above a commutative ring with a one element, then the module is called the dual module of ; the module is called bidual module . There is a canonical mapping ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle M^{*}=\operatorname {Hom} _{A}(M,A)}$ ${\displaystyle M}$ ${\displaystyle M^{**}=\left(M^{*}\right)^{*}}$

${\displaystyle M\to M^{**},\quad m\mapsto (\lambda \mapsto \lambda (m))}$

which is generally neither injective nor surjective. If it is an isomorphism, it is called reflexive. ${\displaystyle M}$

literature

• R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8
• Robert E. Megginson: An Introduction to Banach Space Theory , Springer New York (1998), ISBN 0-387-98431-3 , Chapter 1.3: Characterizations of Reflexivity