A Biased View of Topology as a Tool in Functional Analysis

Abstract

A survey about “Topology as a tool in functional analysis” would be such a giant enterprise that we have, naturally, chosen to give here “Our biased views of topology as a tool in functional analysis”. The consequence of this is that a big portion of this long paper deals with topics that we have been actively working on during the past years. These topics range from metrizability of compact spaces (and their consequences in functional analysis), networks in topological spaces (and their consequences in renorming theory of Banach spaces), distances to spaces of functions (and their applications to the study of pointwise and weak compactness), James’ weak compactness theorem (and their applications to variational problems and risk measures). Some of the results collected here are a few years old while many others are brand new. A few of them are first published here and most of them have been often used in different areas since their publication. The survey is completed with a section devoted to references to some of what we consider the last major achievements in the area in recent years.