9 - Spaces with an “orthogonal” base

Summary

In this chapter we study locally convex spaces E having an “orthogonal” base e₁, e₂, … (9.1.1). We first show that for such E, (weak) sequential completeness, quasicompleteness and completeness are equivalent (9.1.6). E may have closed subspaces and quotients without an “orthogonal” base (9.2.5). We characterize bounded and compactoid sets in E in terms of e₁, e₂, … (9.2.7) and show that compactoids are metrizable (9.2.9(i)). E is semi-Montel if and only if E′_b has an “orthogonal” base (9.2.13). We also characterize semi-Montelness (9.2.15) and nuclearity (9.2.16) in terms of properties of the base.

Every infinite-dimensional Fréchet space contains an infinite-dimensional closed subspace with an “orthogonal” base (9.3.5).

Section 9.4 is a stepping stone for the sequel; here we introduce the perfect sequence spaces and the normal topology in the spirit of the classical spaces of Köthe ([144], 6.30). We prove that it is the class of the spaces E with an “orthogonal” base for which E is weakly sequentially complete and E′ is weakly* sequentially complete (9.4.10).

In Section 9.5 we start with an infinite matrix B of nonnegative real numbers and associate to it a perfect sequence space Λ⁰(B) in a natural way (9.5.2, 9.5.9); we show that the class of these Λ⁰(B) is precisely the class of all Fréchet spaces with an “orthogonal” base (9.5.12). This fact turns out to be very useful; it enables us in Section 9.6 to translate (semi-)Montelness and nuclearity into concrete properties of the matrix B (9.6.2, 9.6.3).We apply this in Section 9.7 to spaces of analytic functions for which we prove properties that have been postponed in previous chapters (9.7.5).