This book is a text for a first course in abstract analysis. Although this topic is traditionally treated in first-year graduate courses, the present volume is so clear and accessible, it is suitable for undergraduates with a good background in the calculus of functions of one and several variables.
In outline, the course consists of a study of the familiar concepts of calculus such as convergence, continuity, differentiation, and integration in a more general and abstract setting. This serves to reinforce and deepen the reader's understanding of the basic concepts of analysis and, at the same time, to provide a familiarity with the abstract approach to analysis which is valuable in many areas of applied mathematics and essential to the study of advanced analysis.
After introductory chapters on sets and relations, the real number system and linear spaces, the book offers easily followed discussions of normed spaces, normed linear spaces, inner product spaces, and a complete development of the Lebesgue integral, as well as important topics involving approximation theory, the Banach fixed-point theorem, and its applications and Stieltjes integrals.
Thorough, well-conceived, and well-written, Real Analysis
is abundantly furnished with problem material—each chapter includes a carefully chosen selection of stimulating problems.