3 - Locally convex spaces

Summary

In this chapter we develop the basics of the main subject of this book, i.e., the theory of locally convex spaces over K. The reader will notice that Sections 3.1–3.6 contain some material that looks familiar to a classical analyst. However, we felt it convenient to give full proofs; it reveals which classical proofs can be translated and what modifications need to be made.

In Section 3.1 we do not immediately consider topologies on our spaces, but introduce seminorms for which we require the strong triangle inequality, and convex sets in an algebraic way. Typical non-Archimedean features here are the solidity of a seminorm (3.1.1) and edged sets (3.1.5, 3.1.13). We prove in 3.1.11 and 3.1.14 that a convex set and a point outside it can be separated by a seminorm (implying that, contrary to the classical situation, convex sets in Kⁿ are closed, 3.4.22(i), 3.4.24).

Section 3.2 is a preparation for Chapter 8 and reading of it may be postponed until that chapter is tackled.

In Section 3.3 we define locally convex spaces in two equivalent ways, one by means of seminorms (3.3.7) and one that requires a neighbourhood base at 0 that consists of convex sets (3.3.16).

In Section 3.4 we consider subspaces (3.4.3), quotients (3.4.6), products (3.4.9), locally convex direct sums (3.4.15), and projective (3.4.29) and inductive (3.4.32) limits of locally convex spaces. We show that every Hausdorff locally convex space can be embedded in a product of Banach spaces (3.4.10), which we use to construct completions, and prove some hereditary properties for completeness.