Real and Functional Analysis

Serge Lang (auth.)9780387940014, 0-387-94001-4

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 Anal­ ysis. I assume that the reader is acquainted with notions of uniform con­ vergence and the like. In this third edition, I have reorganized the book by covering inte­ gration 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 upgrade 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 investiga­ tions, be it in mathematics, or 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.

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Table of contents :
Front Matter….Pages i-xiv
Front Matter….Pages 1-1
Sets….Pages 3-16
Topological Spaces….Pages 17-50
Continuous Functions on Compact Sets….Pages 51-62
Front Matter….Pages 63-63
Banach Spaces….Pages 65-94
Hilbert Space….Pages 95-108
Front Matter….Pages 109-110
The General Integral….Pages 111-180
Duality and Representation Theorems….Pages 181-222
Some Applications of Integration….Pages 223-250
Integration and Measures on Locally Compact Spaces….Pages 251-277
Riemann-Stieltjes Integral and Measure….Pages 278-294
Distributions….Pages 295-307
Integration on Locally Compact Groups….Pages 308-328
Front Matter….Pages 329-329
Differential Calculus….Pages 331-359
Inverse Mappings and Differential Equations….Pages 360-384
Front Matter….Pages 385-386
The Open Mapping Theorem, Factor Spaces, and Duality….Pages 387-399
The Spectrum….Pages 400-414
Compact and Fredholm Operators….Pages 415-437
Spectral Theorem for Bounded Hermitian Operators….Pages 438-463
Further Spectral Theorems….Pages 464-479
Spectral Measures….Pages 480-494
Front Matter….Pages 495-496
Local Integration of Differential Forms….Pages 497-522
Manifolds….Pages 523-546
Integration and Measures on Manifolds….Pages 547-568
Back Matter….Pages 569-580