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, ..., Xn are normed spaces and that X = X1 ⊕ ... ⊕ Xn. Then X is reflexive if and only if each Xj 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.

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