Spaces of Continuous Functions

The space C(X) of all continuous functions on a compact space X carries the structure of a normed vector space, an algebra and a lattice. On the one hand we study the relations between these structures and the topology of X, on the other hand we discuss a number of classical results according to which an algebra or a vector lattice can be represented as a C(X). Various applications of these theorems are given.Some attention is devoted to related theorems, e.g. the Stone Theorem for Boolean algebras and the Riesz Representation Theorem.The book is functional analytic in character. It does not presuppose much knowledge of functional analysis; it contains introductions into subjects such as the weak topology, vector lattices and (some) integration theory.

The book contains many exercises. Some are merely illustrations of the theory, others lead to results used in the text. Solutions to the latter and hints for some of the former are to be found at the back of the book The book considers various disciplines within functional analysis, highlighting connections between, e.g. ordering and multiplication, or Banach algebras and Boolean rings The book has a chapter on spaces of functions with values in other fields that the real or complex numbers Every chapter as an "Extra" of either historical or mathematical character Little knowledge is assumed. There are appendices for the basics of normed vector spaces and lattice theory Includes supplementary material: sn.pub/extras

Inhalt
Topological Preliminaries.- Metrizable Compact Spaces.- The Stone-Weierstrass Theorem.- Weak Topologies. The Alaoglu Theorem.- Riesz Spaces.- Yosida's Representation Theorem.- The Stone-ech compactification.- Evaluations.- C(X) determines X.- The Riesz Representation Theorem.- Banach Algebras.- Other Scalars.