Banach Space - Reflexivity

Reflexivity

If F is also surjective, then the Banach space X is called reflexive. Reflexive spaces have many important geometric properties.

• Theorem Every reflexive normed space is a Banach space.
• Corollary If X is a Banach space, then X is reflexive if and only if X′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.
• Corollary Suppose that X₁, ..., X_n are normed spaces and that X = X₁ ⊕ ... ⊕ X_n. Then X is reflexive if and only if each X_j is reflexive.
• Corollary Let X be a reflexive normed space and Y a Banach space. If there is a bounded linear operator from X onto Y, then Y is reflexive.
• Corollary Let X be a reflexive normed space. Then X is separable if and only if X′ is separable.
• James`s Theorem For a Banach space the following two properties are equivalent:
  • X is reflexive.
  • for all f in X′ there exists x in X with ǁxǁ ≤ 1, so that f(x) = ǁfǁ.
• Lemma A Banach space X is reflexive if and only if the natural pairing on X × X′ is perfect. In particular, X′ is also reflexive then.