Reflexive space

In functional analysis, reflexive spaces are certain Banach spaces which are defined by some abstract property of dual spaces and turn out to have desirable geometric properties.

Table of contents

1 Definition 2 Examples 3 Properties

Definition

Suppose X is a Banach space, and X" its double dual, i.e. the continuous dual space of the continuous dual X' of X. Both X' and X" are Banach spaces, as explained in dual space. 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'.

J is normpreserving (i.e., ||J(x)||=||x|| ) and hence injective as a consequence of the Hahn-Banach theorem. 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.

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 minimial distance between x and C is uniquely defined by x, the point c is not.)

A 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.