Basic Real Analysis

"Basic Real Analysis" demonstrates the richness of real analysis, giving students an introduction both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for example, the construction of real numbers, the Cantor set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. In this modern, systematic text, all the touchstone results and fundamentals are carefully presented, but in a style that requires little prior familiarity with proofs or mathematical language. Basic topics---metric spaces, integration, series, and topology---are methodically developed; advanced material on Banach and Hilbert spaces and Fourier series is also included. Different types of convergence, monotone functions, and applications to probability and other areas of mathematics are featured as well. With its many examples and exercises and broad view of analysis, this work is ideal for senior undergraduates and beginning graduate students, either in the classroom or for self-study.