7 - Barrelledness and reflexivity

Summary

We first follow the “classical” path by developing the notion of barrelledness, a key tool for the theory of reflexivity to be treated in Section 7.4. Indeed, we will prove (7.1.3) that Fréchet spaces are barrelled, by using the Baire Category Theorem.

Nevertheless, for a proper characterization of reflexivity (see 7.4.13) we need a modification of the notion of barrelledness by introducing the wider concept of “polar barrelledness” in 7.1.6. The fact that it is not identical to “ordinary” barrelledness (7.1.10) is a typical non-Archimedean feature. Despite of this, the proofs of the hereditary properties of (polar) barrelledness for quotients (7.1.12(i)), for locally convex direct sums and inductive limits (7.1.13), for products (7.1.15) and completions (7.1.17) are basically “classical”.

In the same spirit we introduce in Section 7.3 the weak star and strong topologies on the dual E′, define reflexivity in 7.4.1 and prove the characterization 7.4.13 of reflexivity. Products and locally convex direct sums of reflexive spaces are reflexive (7.4.23).

However, after this point our results are going to diverge from the classical ones; therefore, we mention a few facts.

For spherically complete K we show in 7.4.19 that every reflexive space is semi-Montel (i.e., each bounded set is a compactoid, see 8.4.1(i) and 8.4.5(δ)), so that reflexive Banach spaces are finite-dimensional. This is in contrast to 7.4.30 showing that every Fréchet space of countable type over a nonspherically complete K is reflexive.

If K is spherically complete, closed subspaces of reflexive Fréchet spaces are reflexive (7.4.27), but if K is not, the reflexive Banach space ℓ^∞ contains a closed non-reflexive subspace (7.4.18(i)).