實分析

《實分析》是2010年8月1日機械工業出版社出版的圖書，作者是羅伊登(Royden.H.L.)。

實分析或實數分析是處理實數及實函數的數學分析。實分析常以基礎集合論，函數概念定義等等開始。其專門研究數列，數列極限，微分，積分及函數序列，以及實函數的連續性，光滑性以及其他相關性質。

Lebesgue Integration for Functions of a Single Real Variable	10 Metric Spaces: Three Fundamental Thanreess
Preliminaries on Sets, Mappings, and Relations	The Arzelb.-Ascoli Theorem
Unions and Intersections of Sets	The Baire Category Theorem
Equivalence Relations, the Axiom of Choice, and Zorn's Lemma	The Banaeh Contraction Principle
1 The Real Numbers: Sets. Sequences, and Functions	H Topological Spaces: General Properties
The Field, Positivity, and Completeness Axioms	Open Sets, Closed Sets, Bases, and Subbases
The Natural and Rational Numbers	The Separation Properties
Countable and Uncountable Sets	Countability and Separability
Open Sets, Closed Sets, and Borel Sets of Real Numbers	Continuous Mappings Between Topological Spaces
Sequences of Real Numbers	Compact Topological Spaces
Continuous Real-Valued Functions of a Real Variable	Connected Topological Spaces
2 Lebesgne Measure	12 Topological Spaces: Three Fundamental Theorems
Introduction	Urysohn's Lemma and the Tietze Extension Theorem
Lebesgue Outer Measure	The Tychonoff Product Theorem
The o'-Algebra of Lebesgue Measurable Sets	The Stone-Weierstrass Theorem
Outer and Inner Approximation of Lebesgue Measurable Sets	13 Continuous Linear Operators Between Bausch Spaces
Countable Additivity, Continuity, and the Borel-Cantelli Lemma	Normed Linear Spaces
Noumeasurable Sets	Linear Operators
The Cantor Set and the Cantor Lebesgue Function	Compactness Lost: Infinite Dimensional Normod Linear Spaces
3 LebesgRe Measurable Functions	The Open Mapping and Closed Graph Theorems
Sums, Products, and Compositions	The Uniform Boundedness Principle
Sequential Pointwise Limits and Simple Approximation	14 Duality for Normed Iinear Spaces
Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem	Linear Ftmctionals, Bounded Linear Functionals, and Weak Topologies
4 Lebesgue Integration	The Hahn-Banach Theorem
The Riemann Integral	Reflexive Banach Spaces and Weak Sequential Convergence
The Lebesgue Integral of a Bounded Measurable Function over a Set of	Locally Convex Topological Vector Spaces
Finite Measure	The Separation of Convex Sets and Mazur's Theorem
The Lebesgue Integral of a Measurable Nonnegative Function	The Krein-Miiman Theorem
The General Lebesgue Integral	15 Compactness Regained: The Weak Topology
Countable Additivity and Continuity of Integration	Alaoglu's Extension of Helley's Theorem
Uniform Integrability: The Vifali Convergence Theorem	Reflexivity and Weak Compactness: Kakutani's Theorem
viii Contents	Compactness and Weak Sequential Compactness: The Eberlein-mulian
5 Lebusgue Integration: Fm'ther Topics	Theorem
Uniform Integrability and Tightness: A General Vitali Convergence Theorem	Memzability of Weak Topologies
Convergence in Measure	16 Continuous Linear Operators on Hilbert Spaces
Characterizations of Riemaun and Lebesgue Integrability	The Inner Product and Orthogonality
6 Differentiation and Integration	The Dual Space and Weak Sequential Convergence
Continuity of Monotone Functions	Bessers Inequality and Orthonormal Bases
Differentiability of Monotone Functions: Lebesgue's Theorem	bAdjoints and Symmetry for Linear Operators
Functions of Bounded Variation: Jordan's Theorem	Compact Operators
Absolutely Continuous Functions	The Hilbert-Schmidt Theorem
Integrating Derivatives: Differentiating Indefinite Integrals	The Riesz-Schauder Theorem: Characterization of Fredholm Operators
Convex Function	Measure and Integration: General Theory
7 The Lp Spaces: Completeness and Appro~umation	17 General Measure Spaces: Their Propertles and Construction
Nor/ned Linear Spaces	Measures and Measurable Sets
The Inequalities of Young, HOlder, and Minkowski	Signed Measures: The Hahn and Jordan Decompositions
Lv Is Complete: The Riesz-Fiseher Theorem	The Caratheodory Measure Induced by an Outer Measure
Approximation and Separability	18 Integration Oeneral Measure Spaces
8 The LP Spacesc Deailty and Weak Convergence	19 Gengral L Spaces:Completeness,Duality and Weak Convergence
The Riesz Representation for the Dual of	20 The Construciton of Particular Measures
Weak Sequential Convergence in Lv	21 Measure and Topbogy
Weak Sequential Compactness	22 Invariant Measures
The Minimization of Convex Functionals	Bibiiography
II Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces	index
9. Metric Spaces: General Properties
Examples of Metric Spaces
Open Sets, Closed Sets, and Convergent Sequences
Continuous Mappings Between Metric Spaces
Complete Metric Spaces
Compact Metric Spaces
Separable Metric Spaces

實分析

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