Basic Real Analysis

One of the bedrocks of any mathematics education, the study of real analysis introduces students both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for instance, the construction of number systems, the Cantor Set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. "Basic Real Analysis" is a modern, systematic text that presents the fundamentals and touchstone results of the subject in full rigor, but in a style that requires little prior familiarity with proofs or mathematical language.Key features include: A broad view of mathematics throughout the book Treatment of all concepts for real numbers first, with extensions to metric spaces later, in a separate chapter Elegant proofs Excellent choice of topics Numerous examples and exercises to enforce methodology; exercises integrated into the main text, as well as at the end of each chapter Emphasis on monotone functions throughout Good development of integration theory Special topics on Banach and Hilbert spaces and Fourier series, often not included in many courses on real analysis Solid preparation for deeper study of functional analysis Chapter on elementary probability Comprehensive bibliography and indexBy covering all the basics and developing rigor simultaneously, this introduction to real analysis is ideal for senior undergraduates and beginning graduate students, both as a classroom text or for self-study. With its wide range of topics and its view of real analysis in a larger context, the book will be appropriatefor more advanced readers as well.

Inhalt
Preface * Set Theory * Sequences and Series of Real Numbers * Limits of Functions * Topology of R and Continuity * Metric Spaces * The Derivative * The Riemann Integral * Sequences and Series of Functions * Normed and Function Spaces * The Lebesgue Integral * Lebesgue Measure * General Measure and Probability * Appendix A: Construction of Real Numbers * References * Index

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