Mathematical Analysis

This is a very good textbook presenting a modern course in analysis both at the advanced undergraduate and at the beginning graduate level. It contains 14 chapters, a bibliography, and an index. At the end of each chapter interesting exercises and historical notes are enclosed.\par From the cover: The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral (of a real-valued function defined on a compact interval). The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean spaces). The final part of the book deals with manifolds, differential forms, and Stokes' theorem [in the spirit of M. Spivak's: Calculus on manifolds (1965; Zbl 141.05403)] which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle . ZENTRALBLATT MATH A. Browder Mathematical Analysis An Introduction Everything needed is clearly defined and formulated, and there is a reasonable number of examples... Anyone teaching a year course at this level to should seriously consider this carefully written book. In the reviewer's opinion, it would be a real pleasure to use this text with such a class. -MATHEMATICAL REVIEWS