Morrison T.J. Functional Analysis - An Introduction to Banach Space Theory

New York: A Wiley-Interscience Publication, JOHN Wiley & SONS INC., 2001. — XII, 361 p.A powerful introduction to one of the most active areas of theoretical and applied mathematics.This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented. While occasionally using the more general topological vector space and locally convex space setting, it emphasizes the development of the reader's mathematical maturity and the ability to both understand and "do" mathematics. In so doing, Functional Analysis provides a strong springboard for further exploration on the wide range of topics the book presents, including:
Weak topologies and applications
Operators on Banach spaces
Bases in Banach spaces
Sequences, series, and geometry in Banach spacesStressing the general techniques underlying the proofs, Functional Analysis also features many exercises for immediate clarification of points under discussion. This thoughtful, well-organized synthesis of the work of those mathematicians who created the discipline of functional analysis as we know it today also provides a rich source of research topics and reference material."A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories; and one can imagine that the ultimate mathematician is one who can see analogies between analogies. " - S. Banach.It is hoped that by studying the ideas and techniques presented in this text, by following through on the directions indicated by many of the fundamental results presented here, and by gaining a deeper understanding of the beauty and subtlety underlying most of Banach's work and legacy, you will develop an appreciation for and understanding of this rich area of mathematics. Much is to be gained from mastering the basic ideas you will be exposed to in the material that follows, and in then continuing to pursue both the theoretical avenues they open and the many applications they represent in mathematics and science. Consider this book a beginning point only."This textbook for a two-semester course in functional analysis presents the basic ideas, techniques, and methods that form the underpinnings of the discipline." "...a useful book which helps the student to understand Banach space theory."Introduction
Basic Definitions and Examples
Basic Principles with Applications
Weak Topologies and Applications
Operators on Banach Spaces
Bases in Banach Spaces
Sequences, Series, and a Little Geometry in Banach Spaces