Groenewegen G.L.M., van Rooij A.C.M. Spaces of Continuous Functions

New York: Atlantis Press, 2016. - 183 p.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.Topological Preliminaries
Metrizable Compact Spaces
The Stone-Weierstrass Theorem
Weak Topologies. The Alaoglu Theorem
Riesz Spaces
Yosida’s Representation Theorem
The Stone-Čech Compactification
Evaluations
C(X) Determines X
The Riesz Representation Theorem
Banach Algebras
Other ScalarsError Report