Reflexive Spaces

Definition (Canonical Embedding)

Let be a normed space. The mapping

where the functional on is defined by

is called the canonical embedding of into its bidual .

The canonical embedding of a normed space into its bidual is well-defined and an embedding of normed spaces.

In particular, is isometric, hence injective.

Proof   We have to show that, for any given , is a bounded linear functional on . Linearity follows from the fact that the vector space structure on is given by pointwise operations. To see that is bounded, observe that

holds for all . Moreover, this implies that . Thanks to Hahn–Banach, we know that there exists a bounded linear functional with such that ; hence, . This means that the mapping is isometric. Clearly, this mapping is also linear, and thus an embedding of normed spaces.

Definition (Reflexivity)

A normed space is said to be reflexive if the canonical embedding into its bidual is surjective.

If a normed space is reflexive, then is isometrically isomorphic with , its bidual. James gives a counterexample for the converse statement.

If a normed space is reflexive, then it is complete; hence a Banach space.

If a normed space is reflexive, then the weak and weak topologies on agree.

Proof   By definition, the weak and weak topologies on are the initial topologies induced by the sets of functionals and , respectively. Since is reflexive, those sets are equal.

The converse is true as well. Proof: TODO

If a normed space is reflexive, then its dual is reflexive.

Proof   Since is reflexive, the canonical embedding

is an isomorphism. Therefore, the dual map

is an isomorphism as well. A priori, it is not clear how is related to the canonical embedding

To show that is surjective, consider any element in . We claim that with . Let be any element of . It is of the form with unique, because is reflexive. We have

by the definition of . On the other hand,

by the definitions of and . This shows that is surjective, hence is reflexive. In fact, we have shown more: .

Every finite-dimensional normed space is reflexive.

Every Hilbert space is reflexive.