Principles of Real AnalysisThe new, Third Edition of this successful text covers the basic theory of integration in a clear, well-organized manner. The authors present an imaginative and highly practical synthesis of the "Daniell method" and the measure theoretic approach. It is the ideal text for undergraduate and first-year graduate courses in real analysis. This edition offers a new chapter on Hilbert Spaces and integrates over 150 new exercises. New and varied examples are included for each chapter. Students will be challenged by the more than 600 exercises. Topics are treated rigorously, illustrated by examples, and offer a clear connection between real and functional analysis. This text can be used in combination with the authors' Problems in Real Analysis, 2nd Edition, also published by Academic Press, which offers complete solutions to all exercises in the Principles text. Key Features: * Gives a unique presentation of integration theory * Over 150 new exercises integrated throughout the text * Presents a new chapter on Hilbert Spaces * Provides a rigorous introduction to measure theory * Illustrated with new and varied examples in each chapter * Introduces topological ideas in a friendly manner * Offers a clear connection between real analysis and functional analysis * Includes brief biographies of mathematicians "All in all, this is a beautiful selection and a masterfully balanced presentation of the fundamentals of contemporary measure and integration theory which can be grasped easily by the student." --J. Lorenz in Zentralblatt für Mathematik "...a clear and precise treatment of the subject. There are many exercises of varying degrees of difficulty. I highly recommend this book for classroom use." --CASPAR GOFFMAN, Department of Mathematics, Purdue University |
Contents
Sequences of Real Numbers | 22 |
The Extended Real Numbers | 29 |
Metric Spaces | 53 |
Separation Properties of Continuous Functions | 80 |
The StoneWeierstrass Approximation Theorem | 87 |
THE THEORY OF MEASURE | 93 |
THE LEBESGUE INTEGRAL | 161 |
NORMED SPACES AND LpSPACES | 217 |
HILBERT SPACES | 275 |
SPECIAL TOPICS IN INTEGRATION | 325 |
Bibliography | 399 |
Common terms and phrases
addition algebra assume ball Banach space Borel set bounded called choose Clearly collection compact complete Consider continuous function converges countable defined Definition denoted dense differentiable disjoint easily easy element equal equicontinuous equivalent establish Example exercise exists extension finite fixed fn(x follows formula function f given hence Hilbert space holds identity implies inequality inner product integrable function interval Lebesgue integrable Lebesgue measure Lemma Let f limit linear functional mathematician measurable function measurable set measurable subset measure space metric space Moreover nonempty norm observe open set operator orthogonal positive preceding Proof properties real numbers real-valued functions regular Borel measure respect result Riemann integrable satisfies semiring sequence Show signed measure step functions Theorem theory topological space uniformly unique vector lattice vector space zero