Reflexive space 

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

Φ : X → X"

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

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 isn't.)

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.

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