Real and Functional Analysis: Third Edition: 142 (Graduate Texts in Mathematics, 142)

This book is meant as a text for a first-year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding, for instance, my undergraduate analysis. I assume that the reader is acquainted with notions of uniform convergence and the like. In this third edition, I have reorganized the book by covering integration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises, as well as some material about integration on the real line, such as Dirac sequence approximation and Fourier analysis, and some material on functional analysis, including the theory of the Gelfand transform in Chapter XVI. These upgrade previous exercises to sections in the text. In a sense, the subject matter covers the same topics as elementary calculus, namely linear algebra, differentiation, and integration. This time, however, these subjects are treated in a manner suitable for the training of professionals, that is, people who will use the tools in further investigations, whether in mathematics, physics, or other fields. In the first part, we begin with point set topology, essential for all analysis, and we cover the most important results.