Functional Analysis for the Practical Man

Abstract

Practical analysis has its origins in geometry. All the theorems of classical analysis and positive results of elementary measure theory are theorems of ZF + DC. Of course, in functional analysis, extensive use has been made of the Axiom of Choice. But, even in functional analysis, many theorems are derived in ZF + DC—for example, the Closed Graph Theorem and the Uniform Boundedness Theorem. Of those theorems, such as the Hahn-Banach Theorem and the Krein-Milman Theorem, whose usual proofs depend on the Axiom of Choice, most can be derived in ZF + DC, provided some mild separability conditions are imposed.