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Exercises in Analysis, Part 1

Leszek Gasiński and Nikolaos S. Papageorgiou
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Problem Books in Mathematics
Problem Book
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Dhruba Adhikari
, on

This is the first of two volumes of Exercises in Analysis in the format of an encyclopedia of problems and their solutions. The book presents most standard theorems in real analysis, topology and functional analysis as well as a variety of problems with their solutions. The topics of main focus are metric spaces, topological spaces with some introductory material in algebraic topology, measure theory and integration (including the interplay between measures and topology), and Banach space theory.

There are five chapters in this volume, each opening with the basic theory and all the main definitions along with theorems. Proofs of the theorems are not presented. This part of the material should help the reader refresh the theory before starting to solve problems. The problems following have complete and detailed solutions; some of them complement proofs of some of the theorems. Over 170 problems for each chapter are given and labeled by the degree of difficulty. The presentation is lucid and elegant. The notations are standard throughout the text.

The selection of problems is carefully done and is in the spirit of what happens in the application of the topics in research fields. The bibliography at the end of each chapter is selected for the benefit of the reader to find related materials in more expanded version and/or in different contexts. Only most widely used and relevant treatises are listed.

The book is useful to graduate students and faculty whose interests are in probability, finance, measure theory, topology, partial differential equations and operator theory; however, it does not seem to be intended for a textbook. The authors have kept the content on the fundamental level so that a diverse group of readers can benefit from it. Even a glance of the solutions of the problems will tell the reader that problem solving skills are vital. A list of other problem books in similar topics is given after the last chapter of the book. Such a book certainly must live in every library where other mathematics books in the similar topics reside.

Dhruba Adhikari is Assistant Professor of Mathematics at Kennesaw State University in Marietta, Georgia, USA.

1. Metric Spaces

2. Topological Spaces

3. Measure, Integral, and Martingales

4. Measures and Topology

5. Functional Analysis

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