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.