Publisher Summary 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.