This text offers an overview of the basic theories and techniques of functional analysis and its applications. It contains topics such as the fixed point theory starting from Ky Fan's KKM covering and quasi-Schwartz operators. It also includes over 200 exercises to reinforce important concepts.;The author explores three fundamental results on Banach spaces, together with Grothendieck's structure theorem for compact sets in Banach spaces (including new proofs for some standard theorems) and Helley's selection theorem. Vector topologies and vector bornologies are examined in parallel, and their internal and external relationships are studied. This volume also presents recent developments on compact and weakly compact operators and operator ideals; and discusses some applications to the important class of Schwartz spaces.;This text is designed for a two-term course on functional analysis for upper-level undergraduate and graduate students in mathematics, mathematical physics, economics and engineering. It may also be used as a self-study guide by researchers in these disciplines.
Table of Contents
Hahn-Banach's Extension Theorem; Banach spaces and Hilbert spaces; operators and some consequences of Hahn-Banach's Extension Theorem; the Uniform Boundedness Theorem and Banach's Open Mapping Theorem; a structure theorem for compact sets in a Banach space; topological injections and topological surjections; vector topologies; separation theorems and Krein-Milman's Theorem; projective topologies and inductive topologies; normed spaces associated with a locally convex space; vector bornologies; initial bornologies and final bornologies; von Neumann bornologies and locally convex topologies determined by convex bornologies; dual pairs and the weak topologies; elementary duality theory; semi-reflexive spaces and ultra-semi-reflexive spaces; some recent results on compact operators and weakly compact operators; precompact seminorms and Schwartz spaces; elementary Riesz - Schauder's theory; an introduction to operator ideals; an aspect of fixed point theory; an inrtoduction to ordered convex spaces.