Reflexive Space - Properties

Properties

If a Banach space Y is isomorphic to a reflexive Banach space X, then Y is reflexive.

Every closed subspace of a reflexive space is reflexive. The dual of a reflexive space is reflexive. Every quotient of a reflexive space is reflexive.

The promised geometric property of reflexive Banach 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. The closest point c is unique when X is uniformly convex.)

Let X be a Banach space. The following are equivalent.

1. The space X is reflexive.
2. The dual of X is reflexive.
3. The closed unit ball of X is compact in the weak topology.(This is known as Kakutani's Theorem.)
4. Every bounded sequence in X has a weakly convergent subsequence.
5. Every continuous linear functional on X attains its maximum on the closed unit ball in X. (James' theorem)

A reflexive Banach space is separable if and only if its dual is separable. This follows from the fact that for every normed space Y, separability of the dual Y ′ implies separability of Y.