基本信息
- 原书名:Real Analysis (4th Edition)
- 原出版社: Prentice Hall
编辑推荐
本书是实分析课程的优秀教材,被国外众多著名大学(如斯坦福大学、哈佛大学等)采用。
内容简介
目录
Preface iii
Lebesgue Integration for Functions of Single Real Variable
Preliminaries on Sets, Mappings, and Relations
UnionsandIntersectionsofSets
Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma .
The Real Numbers: Sets, Sequences, and Functions
1.1 The Field, Positivity, and Completeness Axioms 7
1.2 TheNaturalandRationalNumbers 11
1.3 CountableandUncountableSets . 13
1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers 16
1.5 SequencesofRealNumbers . 20
1.6 Continuous Real-Valued Functions of a Real Variable . 25
Lebesgue Measure 29
2.1 Introduction . 29
2.2 LebesgueOuterMeasure 31
2.3 The σ-AlgebraofLebesgueMeasurableSets . 34
2.4 Outer and Inner Approximation of Lebesgue Measurable Sets 40
2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma . 43
2.6 NonmeasurableSets 47
前言
The book is divided the three parts: Part I treats Lebesgue measure and Lebesgue integration for functions of a single real variable; Part II treats abstract spaces--topological spaces, metric spaces, Banach spaces, and Hilbert spaces; Part III treats integration over general measure spaces, together with the enrichments possessed by the general theory in the presence of topological, algebraic, or dynamical structure.
The material in Parts II and III does not formally depend on Part I. However, a careful treatment of Part I provides the student with the opportunity to encounter new concepts in a familiar setting, which provides a foundation and motivation for the more abstract concepts developed in the second and third parts. Moreover, the Banach spaces created in Part I, the LP spaces, are one of the most important classes of Banach spaces. The principal reason for establishing the completeness of the LP spaces and the characterization of their dual spaces is to be able to apply the standard tools of functional analysis in the study of functionals and operators on these spaces. The creation of these tools is the goal of Part II.
NEW TO THE EDITION
·This edition contains 50% more exercises than the previous edition
·Fundamental results, including Egoroff's Theorem and Urysohn's Lemma are now proven in the text.
·The Borel-Cantelli Lemma, Chebychev's Inequality, rapidly Cauchy sequences, and the continuity properties possessed both by measure and the integral are now formally presented in the text along with several other concepts.
There are several changes to each part of the book that are also noteworthy:
Part I
·The concept of uniform integrability and the Vitali Convergence Theorem are now presented and make the centerpiece of the proof of the fundamental theorem of integral calculus for the Lebesgue integral
·A precise analysis of the properties of rapidly Cauchy sequences in the LP(E) spaces, 1 ≤ p ≤ ∞, is now the basis of the proof of the completeness of these spaces
·Weak sequential compactness in the LP(E) spaces, 1≤p≤∞, is now examined in detail and used to prove the existence of minimizers for continuous convex functionals.
Part II
·General structural properties of metric and topological spaces are now separated into two brief chapters in which the principal theorems are proven.
·In the treatment of Banach spaces, beyond the basic results on bounded linear operators, compactness for weak topologies induced by the duality between a Banach space and its dual is now examined in detail.
·There is a new chapter on operators in Hilbert spaces, in which weak sequential compactness is the basis of the proofs of the Hilbert-Schmidt theorem on the eigenvectors of a compact symmetric operator and the characterization by Riesz and Schuader of linear Fredholm operators of index zero acting in a Hilbert space.
Part III
·General measure theory and general integration theory are developed, including the completeness, and the representation of the dual spaces, of the LP(X, μ) spaces for, 1 ≤ p ≤ ∞. Weak sequential compactness is explored in these spaces, including the proof of the Dunford-Pettis theorem that characterizes weak sequential compactness in L1 (X, μ).
·The relationship between topology and measure is examined in order to characterize the dual of C(X), for a compact Hausdorff space X. This leads, via compactness arguments, to (i) a proof of yon Neumann's theorem on the existence of unique invariant measures on a compact group and (ii) a proof of the existence, for a mapping on a compact Hausdorf space, of a probability measure with respect to which the mapping is ergodic.
The general theory of measure and integration was born in the early twentieth century. It is now an indispensable ingredient in remarkably diverse areas of mathematics, including probability theory, partial differential equations, functional analysis, harmonic analysis, and dynamical systems. Indeed, it has become a unifying concept. Many different topics can agreeably accompany a treatment of this theory. The companionship between integration and functional analysis and, in particular, between integration and weak convergence, has been fostered here: this is important, for instance, in the analysis of nonlinear partial differential equations (see L.C. Evans' book Weak Convergence Methods for Nonlinear Partial Differential Equations [AMS, 1998]).
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