Real and Functional Analysis (3rd Edition)

Author: S.Lang
Publisher:
Publish Date: 1997-09-01
Features: Real and Functional Analysis (3rd Edition, English Edition) is published by World Scientific Publishing Company.
Foreword
This book is intended 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 class. 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 (e.g., on Dirac sequence approximation and on Fourier analysis), and some material on functional analysis (e.g., the theory of the Gelfand transform in Chapter XVI). These upgrades raise the previous exercises to sections in the text. In a sense, the subject matter covers the same topics as elementary calculus, viz. linear algebra, differentiation, and integration. This time, however, these subjects are treated in a manner suitable for the training of professionals, i.e., people who will use the tools in further investigations, be it in mathematics, physics, or what have you.
In the first part, we begin with point-set topology, essential for all analysis, and we cover the most important results. I am selective here, since this part is regarded as a tool, especially Chapters 1 and II. Many results are easy and are less essential than those in the text. They have been given in exercises, which are designed to acquire facility in routine techniques and to give flexibility for those who want to cover some of them in greater length. The point-set topology simply deals with the basic notions of continuity, open and closed sets, connectedness, compactness, and continuous functions. The chapter concerning continuous functions on compact sets properly emphasizes results that already mix analysis and uniform convergence with the language of point-set topology.
In the second part, Chapters IV and V, we describe briefly the two basic linear spaces of analysis, namely Banach spaces and Hilbert spaces. The next part deals extensively with integration. We begin with the development of the integral. The fashion has been to emphasize positivity and ordering properties (increasing and decreasing sequences). I find this excessive. The treatment given here attempts to give a proper balance between L1-convergence and positivity. For more detailed comments, see the introduction to Part Three and Chapter VI. The chapters on applications of integration and distributions provide concrete examples and choices for leading the course in other directions, at the taste of the lecturer. The general theory of integration in measured spaces (with respect to a given positive measure) alternates with chapters giving specific results of integration on Euclidean spaces or the real line. Neither is slighted at the expense of the other.
In this third edition, I have added some material on functions of bounded variation, and I have emphasized convolutions and the approximation by Dirac sequences or families even more than in the previous editions, for instance, in Chapter VIII. For want of a better place, the calculus (with values in a Banach space) now occurs as a separate part after dealing with integration and before functional analysis. The differential calculus is done because, at best, most people will only be acquainted with it only in Euclidean space, and incompletely at that. More importantly, the calculus in Banach spaces has acquired considerable importance in the last two decades, because of many applications like Morse theory, the calculus of variations, and the Nash-Moser implicit mapping theorem, which lies even further in this direction since one has to deal with more general spaces than Banach spaces. These results pertain to the geometry of function spaces. Cf. the exercises of Chapter XIV for simpler applications.
The next part deals with functional analysis. The purpose here is twofold. We place the linear algebra in an infinite-dimensional setting where continuity assumptions are made on the linear maps, and we show how one can "linearize" a problem by taking derivatives, again in a setting where the theory can be applied to function spaces. This part includes several major spectral theorems of analysis, showing how we can extend to the infinite dimension...