Reflexive space

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This page concerns the reflexivity of a Banach space. For Paul Halmos' notion of the reflexivity of an operator algebra or a subspace lattice, see reflexive operator algebra.

In functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving dual spaces. Reflexive spaces turn out to have desirable geometric properties.

Table of contents

1 Definition
2 Examples
3 Properties
4 Implications

Definition

Suppose X is a Banach space. We denote by X' its continuous dual, i.e. the space of all continuous linear maps from X to the base field (R or C). This is again a Banach space, as explained in the dual space article. So we can form the double dual X", the continuous dual of X'. There is a natural continuous linear transformation

J : X → X"

defined by

J(x)(φ) = φ(x) for every x in X and φ in X'.

As a consequence of the Hahn-Banach theorem, J is norm-preserving (i.e., ||J(x)||=||x|| ) and hence injective. The space X is called reflexive if J is bijective.

Examples

All Hilbert spaces are reflexive, as are the L^p spaces for 1 < p < ∞. More generally: all uniformly convex Banach spaces are reflexive according to the Milman-Pettis theorem.

Properties

Every closed subspace of a reflexive space is reflexive.

The promised geometric property of reflexive spaces is the following: if C is a closed non-empty convex subset of the reflexive space X, then for every x in X there exists a c in C such that ||x - c|| minimizes the distance between x and points of C. (Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.)

A banach space is reflexive if and only if its dual is reflexive.

A space is reflexive if and only if its unit ball is compact in the weak topology.

Implications

A reflexive space is separable if and only if its dual is separable.

If a space is reflexive, then every bounded sequence has a weakly convergent subsequence.