# Mathematical Analysis I

Elias Zakon

Copyright Year: 2004

ISBN 13: 978-1-931705-02-X

Publisher: Independent

Language: English

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## Reviews

This first volume offers an excellent introduction to analysis. There is easily sufficient material for a two-semester undergraduate course beginning with ideas from set theory and the real numbers, the author moves immediately to vector spaces... read more

This first volume offers an excellent introduction to analysis. There is easily sufficient material for a two-semester undergraduate course beginning with ideas from set theory and the real numbers, the author moves immediately to vector spaces and metric spaces; in particular, covering both E^n and C^n. Once the properties of these spaces have been covered the text advances to functions, limits, and continuity while also covering basic topological ideas such as compactness and connectedness. Infinite sequences and series, including power series, are also discussed. Finally, differentiation and antidifferentiation are tackled. Topics here include L'Hopital's Rule, Taylor's series, rectifiable curves, and integral definitions of certain functions (e.g., ln(x) and the inverse trigonometric functions). More advanced topics, such as the theory of metric spaces, Riemann-Stieltjes integration, and Lebesgue theory are not covered in this volume, but are reserved for the author's Mathematical Analysis II. The index is very detailed and the entries are hyperlinked to the text.

This book appears to be quite well-written and error-free.

Mathematical analysis is a cornerstone of mathematics. As such, the content of this book is highly relevant to any mathematical scientist. The text provides a solid foundation for students of mathematics, physics, chemistry, or engineering. The level of depth and rigor is appropriate for an undergraduate audience and would form a solid basis for future study at the graduate level.

Originally published in 1975, this is the one area in which the text shows its age. The book leans heavily on the notation of symbolic logic. Modern students may require some extra assistance in parsing some the notation in the statements of definitions, theorems, and problems. To his credit, the author often explains in words a definition or theorem before giving the formal statement using the symbolism.

The vocabulary and notation used in this text is standard. The author maintains consistent notation and vocabulary throughout the text making it easy to transition from one topic to the next. The author does extensively utilize the notation of symbolic logic; however, this notation is introduced in the very first chapter during a review of the basics of set theory.

The book is broken into sections of reasonable length each covering a particular topic. Problem sets follow every few sections and are focused on the content of those sections. As is true for most mathematics texts, this book should be treated in a sequential manner; however, the author does note those sections that can be omitted without a loss of continuity. Also, with appropriate development of notation, individual chapters of this book could be used in conjunction with or to supplement another text.

The text is organized in five chapters each having several chapters (from nine to seventeen). The book is designed to be covered in the order written, though the author does indicate those sections that can (or even should) be omitted from an introductory course. The topics are developed in a logical order as the text builds later concepts from earlier ideas in a traditional manner.

The presentation and typesetting of formulas, equations, etc. is very well-done. The figures, graphs, and images are clear and integrated into the text. References to earlier results as well as entries in the index are hyperlinked to allow the reader to easily navigate the text.

I found no grammatical errors in the text.

Since this is a book on abstract mathematics, there are no cultural references in the text.

## Table of Contents

- Chapter 1. Set Theory
- Chapter 2. Real Numbers. Fields
- Chapter 3. Vector Spaces. Metric Spaces
- Chapter 4. Function Limits and Continuity
- Chapter 5. Differentiation and Antidifferentiation

## About the Book

This award-winning text carefully leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor's theorem, total variation, rectifiable arcs, and sufficient conditions of integrability. Well over 500 exercises (many with extensive hints) assist students through the material.

For students who need a review of basic mathematical concepts before beginning "epsilon-delta"-style proofs, the text begins with material on set theory (sets, quantifiers, relations and mappings, countable sets), the real numbers (axioms, natural numbers, induction, consequences of the completeness axiom), and Euclidean and vector spaces; this material is condensed from the author's Basic Concepts of Mathematics, the complete version of which can be used as supplementary background material for the present text.

## About the Contributors

### Author

**Elias Zakon, **As a research fellow at the University of Toronto, he worked with Abraham Robinson. In 1957, he joined the mathematics faculty at the University of Windsor, where the first degrees in the newly established Honours program in Mathematics were awarded in1960. While at Windsor, he continued publishing his research results in logic and analysis. In this post-McCarthy era, he often had as his house-guest the prolific and eccentric mathematician Paul Erdos, who was then banned from the United States for his political views. Erdos would speak at the University of Windsor, where mathematicians from the University of Michigan and other American universities would gather to hear him and to discuss mathematics. While at Windsor, Zakon developed three volumes on mathematical analysis,which were bound and distributed to students. His goal was to introduce rigorous material as early as possible; later courses could then rely on this material. We are publishing here the latest complete version of the second of these volumes, which was used in a two-semester class required of all second-year Honours Mathematics students at Windsor.