Introduction to Real Analysis

by William F. Trench

Using an extremely clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. The real number system. Differential calculus of functions of one variable. Riemann integral functions of one variable. Integral calculus of real-valued functions. Metric Spaces. For those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

Genres: Textbooks, Mathematics

574 pages, Hardcover

First published December 4, 2002

Reviews

[Matt · Rating 4 out of 5]

Despite some typos and miniscule errors, and the author's differing choice of symbolism (e.g. how he denotes complements, boundaries, closures, etc.), this is a good and expansive textbook on analysis for both single variable, real-valued functions, sequences and series as well as multivariable, vector-valued functions in addition to metric spaces.

Trench recommends this as a two-semester textbook, although I'd say a prior class in logic and analysis is strongly encouraged before engaging this material, as many basics including methods of proof and the elementary concepts and theorems for the real line (e.g. Heine-Borel, Bolzano-Weierstrass) are covered very quickly. Also, his choice to cover sequences and series after derivatives and integrals seems counterintuitive, since most books cover them in the opposite order.

The chapters on real-valued and vector-valued functions defined over multi-dimensional spaces, together with a rigorous treatment of the inverse function and implicit function theorems, are a definite advantage over other texts, and the whole chapter on metric spaces including bridges to function spaces and fixed point theorems is particularly handy for economics majors.

Overall, 4 out of 5.