CHAPTER 23 - Banach Spaces and Topology

Abstract

The theory of Banach spaces has its foundation on set-theoretic topology. Thus, the classical work of Banach rests heavily on the deeper properties of topology: completeness of the reals gives the Hahn–Banach theorem. Baire's category theorem proves the open mapping theorem and the uniform boundedness principle, while Tychonoff's compactness theorem proves the Alaoglou theorem. The interaction between set-theoretic topology and Banach spaces is not only internal, that is, in the sense of using analytic methods in the study of topological spaces and conversely of using topological arguments in the study of Banach spaces. The interaction is to a large extent also because of the fact that both branches use the same set-theoretic methods Set-theoretic, topological, and Banach spaces problems and methods coalesce and interact in problems concerning calibers, existence of independent families, and some isomorphic embeddings into Banach spaces. This chapter further discusses infinitary combinatorics. The ordinals are defined in such a way that an ordinal is the set of smaller ordinals.