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.

chapter 1|12 pages

Hahn-Banach’s Extension Theorem

chapter 2|34 pages

Banach Spaces and Hilbert Spaces

chapter 7|22 pages

Vector Topologies

chapter 13|10 pages

Vector Bornologies

chapter 14|8 pages

Initial Bornologies and Final Bornologies

chapter 16|11 pages

Dual Pairs and the Weak Topology

chapter 17|22 pages

Elementary Duality Theory

chapter 20|17 pages

Precompact Seminorms and Schwartz Spaces

chapter 21|15 pages

Elementary Riesz-Schauder’s Theory

chapter 22|34 pages

An Introduction to Operator Ideals

chapter 24|34 pages

An Introduction to Ordered Convex Spaces