Real and functional analysis. 3. ed. (English) Zbl 0831.46001

Lang, Serge

Graduate Texts in Mathematics. 142. New York: Springer-Verlag. xiv, 580 p. (1993).

This classic textbook [formerly named “Analysis. II” (1969; Zbl 0176.005) and “Real Analysis” (1984; Zbl 0502.46003)] now appears in a 3rd edition, whose new title reflects the contents best. Since the former editions are well known, let us only cite the paragraph with the description of the changes in the author’s preface:
“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 upgrade previous exercises to sections in the text.
In my mind, this is still the best textbook on the market”.

Reviewer: J.Lorenz (Berlin)

MSC:

46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46G05	Derivatives of functions in infinite-dimensional spaces
28A25	Integration with respect to measures and other set functions
28-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
46F10	Operations with distributions and generalized functions
46G10	Vector-valued measures and integration
28B05	Vector-valued set functions, measures and integrals
46B10	Duality and reflexivity in normed linear and Banach spaces
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
43A05	Measures on groups and semigroups, etc.
47A10	Spectrum, resolvent
47A53	(Semi-) Fredholm operators; index theories
47B07	Linear operators defined by compactness properties
47B15	Hermitian and normal operators (spectral measures, functional calculus, etc.)
26B20	Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
58C35	Integration on manifolds; measures on manifolds

Keywords:

Banach and Hilbert spaces; operator and spectral theory; spectral measures; distributions; integration of vector functions; Haar measure; Fourier integral; differential calculus on Banach spaces and manifolds; flows; differential forms; Stokes theorem with singularities; harmonic analysis; vector measures; Dirac sequence approximation; Fourier analysis; Gelfand transform

Citations:

Zbl 0176.005; Zbl 0502.46003