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.

Aim

This book presents the basics of locally convex theory over a field K with a non-Archimedean valuation ∣.∣ : K →[0,∞) (see 1.2.3). The most important example of such a K is the field of the p-adic numbers (1.2.7). The strong triangle inequality ∣λ + μ∣ ≤ max(∣λ∣,∣μ∣) is the major difference between ∣.∣ and the absolute value function on the field of real numbers ℝ and the field of complex numbers ℂ. Likewise, the defining seminorms of our locally convex spaces will satisfy the strong triangle inequality.

The book is self-contained in the sense that it does not require knowledge of any deep theory; only basic knowledge of (linear) algebra, analysis and topology are needed. It is intended for both (graduate) students and interested researchers in other areas, but is also of relevance for specialists.

History

The founding father of non-Archimedean Functional Analysis was Monna, who wrote a series of papers in 1943 (see [152]–[155]). Over the years a wellestablished discipline developed, reflected in the 2000 Mathematics Subject Classifications 46S10 and 47S10 of the Mathematical Reviews. A milestone was reached in 1978 at the publication of van Rooij's book [193], themost extensive treatment on non-Archimedean Banach spaces existing in the literature. In the meantime van Tiel had published his thesis [227] on non-Archimedean locally convex spaces. Both fundamental works still form a basis for new developments, and have been cited by many authors.