zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A course in real analysis. Biographies by Carol A. Weiss. (English) Zbl 0965.26001
San Diego, CA: Academic Press. xx, 745 p. £49.95; $ 69.95 (1999).

This is a book about real analysis, but it is not an ordinary real analysis book. Written with the student in mind, this text incorporates pedagogical techniques not often found in books at this level. The book is intended for a one-year course in real analysis at the graduate level or the advanced undergraduate level. The text material has been class tested several times and has been used for independent study courses as well. This book contains many features that are unique for a real analysis text. Here are a few. Motivation of key concepts. Detailed theoretical discussion. Illustrative examples. Abundant and varied exercises. Applications. The text offers considerable flexibility in the choice of material to cover.

Chapters 1 and 2 (set theory, real numbers, and calculus) present prerequisite material that provides a common ground for all readers. Chapters 3 and 4 present the elements of measure and integration by first discussing the Lebesgue theory on the line (Chapter 3) and then the abstract theory (Chapter 4). This material is prerequisite to all subsequent chapters. Chapter 5 provides an introduction to the fundamentals of probability theory, including the mathematical model for probability, random variables, expectation, and laws of large numbers. In Chapter 6 differentiation is discussed, both of functions and of measures. Topics examined include differentiability, bounded variation, and absolute continuity of functions, and a thorough discussion of signed and complex measures, the Radon-Nikodým theorem, decomposition of measures, and measurable transformations. Chapter 7 provides the fundamentals of topological and metric spaces. In addition to topics traditionally found in an introduction to topology, a discussion of weak topologies and function spaces is included. Completeness, compactness, and approximation comprise the topics for Chapter 8. Examined therein are the Baire category theorem, contractions of complete metric spaces, compactness in function and product spaces, and the Stone-Weierstrass theorem. Presented in Chapter 9 are Hilbert spaces and the classical Banach spaces. Among other things, bases and duality in Hilbert space, completeness and duality of p -spaces, and duality in spaces of continuous functions are discussed. The basic theory of normed and locally convex spaces is given in Chapter 10. Topics include the Hahn-Banach theorem, linear operators on Banach spaces, fundamental properties of locally convex spaces, and the Krein-Milman theorem. Chapter 11 provides applications of previous chapters to harmonic analysis. We examine the elements of Fourier series and transforms and the 2 -theory of the Fourier transform. In addition, an introduction to wavelets and the wavelet transform is presented. Chapter 12 examines measurable dynamical systems. This chapter requires the one on probability (Chapter 5) and discusses ergodic theorems, isomorphisms of measurable dynamical systems, and entropy.


MSC:
26-01Textbooks (real functions)
00A05General mathematics
28-01Textbooks (measure and integration)
46-01Textbooks (functional analysis)
54-01Textbooks (general topology)