S. Kesavan – författare

This book is suited for a first course on Functional Analysis at the masters level. Efforts have been made to illustrate the use of various results via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. So, this book will be particularly useful for students who aspire to a research career in the applications of mathematics.Special emphasis has been given to the treatment of weak topologies and their applications to notions like reflexivity, separability and uniform convexity. The chapter on Lebesgue spaces includes a section devoted to the simplest examples of Sobolev spaces. The chapter on compact operators includes the spectral theory of compact self-adjoint operators on a Hilbert space.Each chapter concludes with a large selection of exercises of varying degrees of difficulty. They often provide examples or counter-examples to illustrate the optimality of the hypotheses of various theorems proved in the text, or develop simple versions of theories not developed therein.In this (second) edition, the book has been completely overhauled, without altering its original structure. Proofs of many results have been rewritten for greater clarity of exposition. Many examples have been added to make the text more user-friendly. Several new exercises have been added.

This book deals with topics usually studied in a masters or graduate level course on the theory of measure and integration. It starts with the Riemann integral and points out some of its shortcomings which motivate the theory of measure and the Lebesgue integral.Starting with abstract measures and outermeasures, the Lebesgue measure is constructed and its important properties are highlighted. Measurable functions, different notions of convergence, the Lebesgue integral, the fundamental theorem of calculus, product spaces, and signed measures are studied. There is a separate chapter on the change of variable formula and one on Lp- spaces. Most of the material in this book can be covered in a one semester course. The prerequisite for following this book is familiarity with basic real analysis and elementary topological notions, with special emphasis on the topology of the N- dimensional euclidean space. Each chapter is provided with a variety of exercises.

This second edition is thoroughly revised and includes several new examples and exercises. Proofs of many results have been rewritten for a greater clarity. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of the topics to study differential equations and calculus of variations. The book includes a chapter on weak topologies and their applications. It also includes a chapter on the Lebesgue spaces, which discusses Sobolev spaces. The book includes a chapter on compact operators and their spectra, especially for compact self-adjoint operators on a Hilbert space. Each chapter has a large collection of exercises in the end, which give additional examples and counterexamples to the results given in the text. This book is suitable for a first course in functional analysis for graduate students who wish to pursue a career in the applications of mathematics.

The study of isoperimetric inequalities involves a fascinating interplay of analysis, geometry and the theory of partial differential equations. Several conjectures have been made and while many have been resolved, a large number still remain open.One of the principal tools in the study of isoperimetric problems, especially when spherical symmetry is involved, is Schwarz symmetrization, which is also known as the spherically symmetric and decreasing rearrangement of functions. The aim of this book is to give an introduction to the theory of Schwarz symmetrization and study some of its applications.The book gives an modern and up-to-date treatment of the subject and includes several new results proved recently. Effort has been made to keep the exposition as simple and self-contained as possible. A knowledge of the existence theory of weak solutions of elliptic partial differential equations in Sobolev spaces is, however, assumed. Apart from this and a general mathematical maturity at the graduate level, there are no other prerequisites.