Analysis (Graduate Studies in Mathematics, Vol. 14)

Noted authorities Elliott Lieb and Michael Loss take readers quickly from basic topics to practical applications, incorporating only those results and constructions that work successfully in mathematics and its applications, while omitting typical textbook topics usually included for historical reasons or to achieve (sometimes unneeded) generality. Analysis includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to a new level of understanding with a minimum of fuss, while at the same time doing so in a rigorous and pedagogical way.

The authors take great care to include topics that any working analyst uses in everyday practice. Unlike other books on the topic, Analysis does not try to laboriously develop subjects from basic principles to full generality. Instead, the book immediately starts working with a subject and shows its relation to other mathematical concepts, illustrating how the corresponding notions work, how to apply them in practice, and more.

The book covers measure and integration, theory of $L^p$ spaces, distribution theory, Fourier analysis, potential theory, and Sobolev spaces. Certain less standard topics (such as rearrangement, integral, and Sobolev inequalities) are included to give the reader a flavor of research in analysis and to emphasize the open, active nature of this area. To illustrate the use of mathematics developed in the book, the concluding chapter contains three examples of solving problems from the calculus of variations.

Analysis is a unique, practical book that everyone--from the graduate student, to the professional mathematician, to the physicist or engineer using analytical methods--will find interesting, stimulating, and useful.