Real Analysis, 4/e (Paperback)
暫譯: 實變函數論 (第4版)

Halsey L. Royden , Patrick M. Fitzpatrick

Real Analysis, 4/e (Paperback)

商品描述

本書序言

●50% more exercises than the previous edition gives students the practice they need to learn and master the material. The exercises range from those that confirm understanding of fundamental ideas and results to those that offer significant mathematical challenge; many exercises foreshadow future developments.
●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.

本書特色

●Independent, modular chapters give instructors the freedom to arrange the material into a course according that suits their needs. A chart in the text gives the essential dependencies.
●Content is divided into 3 parts: Part 1, Classical theory of functions, including the classical Banach spaces; Part 2, General topology and the theory of general Banach spaces; Part 3, Abstract treatment of measure and integration.
。Part 1 is a thorough presentation of Lebesgue measure on the real line and the Lebesgue integral for functions of a single variable. Detailed proofs of all major results are now presented in the text. The concept of uniform inte­grability is now prominently placed. An examination of the concept of weak convergence in the Lp spaces, with an applications to the minimization of con­vex functionals, concludes the first part.
。Part 2 is now a significantly expanded presentation of abstract spaces: metric, topological, Banach, and Hilbert. Foun­dational results for metric spaces (the Baire Category Theorem), for topological spaces (Urysohn's Lemma and the Tychonoff Product Theorem), and for linear spaces (the Hahn-Banach Theorem) are established and employed to create such basic tools for the analysis of linear operators and functionals as the Open Map­ping Theorem, the Uniform Boundedness Principal, Alaoglu's Theorem, and the Krein-Milman Theorem.
。Part 3 starts with a presentation of the basic theory of general measure spaces and integration over such spaces, in the absence of any topological or algebraic structure. Lebesgue measure on Euclidean space is examined. Product measures are examined, the main result being Fubini's Theorem. Several selected topics are then explored.

商品描述(中文翻譯)

本書序言

●本書比前一版多出50%的練習題，為學生提供了學習和掌握材料所需的練習。這些練習題從確認基本概念和結果的理解到提供顯著的數學挑戰不等；許多練習題預示著未來的發展。

●基本結果，包括Egoroff定理和Urysohn引理，現在在文本中得到了證明。

●Borel-Cantelli引理、切比雪夫不等式、快速Cauchy序列，以及測度和積分所具備的連續性特性現在在文本中正式呈現，還有幾個其他概念。

本書特色

●獨立的模組化章節使得教師可以自由地根據自己的需求安排課程內容。文本中的圖表提供了基本的依賴關係。

●內容分為三個部分：第一部分，函數的古典理論，包括古典Banach空間；第二部分，通用拓撲學和通用Banach空間的理論；第三部分，測度和積分的抽象處理。

。第一部分徹底介紹了實數線上的Lebesgue測度和單變數函數的Lebesgue積分。所有主要結果的詳細證明現在在文本中呈現。均勻可積性概念現在被突出展示。對Lp空間中弱收斂概念的檢視，以及對凸泛函最小化的應用，結束了第一部分。

。第二部分現在是對抽象空間的顯著擴展介紹：度量空間、拓撲空間、Banach空間和Hilbert空間。度量空間的基礎結果（Baire類別定理）、拓撲空間的基礎結果（Urysohn引理和Tychonoff乘積定理）以及線性空間的基礎結果（Hahn-Banach定理）被建立並用來創建分析線性算子和泛函的基本工具，如開映射定理、均勻有界原理、Alaoglu定理和Krein-Milman定理。

。第三部分以一般測度空間的基本理論和在這些空間上進行積分的介紹開始，無需任何拓撲或代數結構。對歐幾里得空間上的Lebesgue測度進行了檢視。對乘積測度進行了研究，主要結果是Fubini定理。然後探討幾個選定的主題。

目錄大綱

PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE
1. The Real Numbers: Sets, Sequences and Functions
2. Lebesgue Measure
3. Lebesgue Measurable Functions
4. Lebesgue Integration
5. Lebesgue Integration: Further Topics
6. Differentiation and Integration
7. The LΡ Spaces: Completeness and Approximation
8. The LΡ Spaces: Duality and Weak Convergence
PART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT
9. Metric Spaces: General Properties
10. Metric Spaces: Three Fundamental Theorems
11. Topological Spaces: General Properties
12. Topological Spaces: Three Fundamental Theorems
13. Continuous Linear Operators Between Banach Spaces
14. Duality for Normed Linear Spaces
15. Compactness Regained: The Weak Topology
16. Continuous Linear Operators on Hilbert Spaces
PART III: MEASURE AND INTEGRATION: GENERAL THEORY
17. General Measure Spaces: Their Properties and Construction
18. Integration Over General Measure Spaces
19. General LΡ Spaces: Completeness, Duality and Weak Convergence
20. The Construction of Particular Measures
21. Measure and Topology
22. Invariant Measures

目錄大綱(中文翻譯)

PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE

1. The Real Numbers: Sets, Sequences and Functions

2. Lebesgue Measure

3. Lebesgue Measurable Functions

4. Lebesgue Integration

5. Lebesgue Integration: Further Topics

6. Differentiation and Integration

7. The LΡ Spaces: Completeness and Approximation

8. The LΡ Spaces: Duality and Weak Convergence

PART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT

9. Metric Spaces: General Properties

10. Metric Spaces: Three Fundamental Theorems

11. Topological Spaces: General Properties

12. Topological Spaces: Three Fundamental Theorems

13. Continuous Linear Operators Between Banach Spaces

14. Duality for Normed Linear Spaces

15. Compactness Regained: The Weak Topology

16. Continuous Linear Operators on Hilbert Spaces

PART III: MEASURE AND INTEGRATION: GENERAL THEORY

17. General Measure Spaces: Their Properties and Construction

18. Integration Over General Measure Spaces

19. General LΡ Spaces: Completeness, Duality and Weak Convergence

20. The Construction of Particular Measures

21. Measure and Topology

22. Invariant Measures

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