Mathematical Analysis I – 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.
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
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About the Contributors
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.