【2015新书】An Introduction to Modern Analysis

Book 图书名称：An Introduction to Modern Analysis
Author 作者：Vicente Montesinos, Peter Zizler, Václav Zizler
Publisher 出版社： Springer
Page 页数：863
Publishing Date 出版时间：Jun 14, 2015                        
Language 语言：English
Size 大小：9.7 MB
Format 格式：pdf  文字版
ISBN:3319124803, 9783319124803, 9783319124810
Edition:   第1版  搜索过论坛，没有该文档


Carefully examines the main principles, results and techniques in advanced undergraduate real analysis courses
Fully self-contained, it presents proofs and an ample amount of nontrivial exercises with hints to help to master the subject
Provides links to several areas of modern analysis like Functional analysis, Fourier analysis and Nonlinear analysis at the graduate level
Individual chapters may be downloaded separately for professors interested in teaching a particular topic in-depth

Examining the basic principles in real analysis and their applications, this text provides a self-contained resource for graduate and advanced undergraduate courses. It contains independent chapters aimed at various fields of application, enhanced by highly advanced graphics and results explained and supplemented with practical and theoretical exercises. The presentation of the book is meant to provide natural connections to classical fields of applications such as Fourier analysis or statistics. However, the book also covers modern areas of research, including new and seminal results in the area of functional analysis.

== Table of contents ==
Table of contents :
List of Figures

List of Tables


Chapter 1 Real Numbers: The Basics
1.1 Notation
1.2 Natural Numbers
1.3 Integers
1.4 Fractions and Rational Numbers
1.4.1 Introduction
1.4.2 Powers and Radicals of Rational Numbers
1.5 Base Representation
1.5.1 The Expansion of a Natural Number in Base b
1.5.2 The Expansion of a Rational Number in Base b
1.6 Real Numbers
1.6.1 The Definition of a Real Number
1.6.2 The Expansion of a Real Number in Base b.
1.6.3 The Extended Real Number System, Intervals
1.6.4 Order Properties---and the Completeness---of R
1.7 Cardinality of Sets
1.7.1 Basics on Cardinality
1.7.2 Cardinality of Z and Q
1.7.3 Cardinality of R
1.7.4 Cardinality of the Set of Real Functions
1.8 Topology of R
1.8.1 Introduction. Open and Closed Sets
1.8.2 Neighborhoods, Closure, Interior
1.8.3 Topology on a Subset
1.8.4 Compactness
1.8.5 Connectedness and Related Concepts
1.9 The Baire Category Theorem in R

Chapter 2 Sequences and Series
2.1 Approximation by Rational Numbers
2.2 Sequences
2.2.1 Basics on Sequences
2.2.2 Two Particular Sequences: Arithmetic and Geometric Progressions
Arithmetic Progressions
Geometric Progressions
2.3 More on Sequences
2.4 Series
2.4.1 Introduction
2.4.2 General Criteria for Convergence of Series
2.4.3 Series of Nonnegative Terms
2.4.4 Series of Arbitrary Terms
2.4.5 Rearrangement of Series
2.4.6 Double Sequences and Double Series
Double Sequences
Double Series
2.4.7 Product of Series
2.5 The Euler Number e
2.6 Infinite Products

Chapter 3 Measure
3.1 Measure
3.1.1 The Lebesgue Outer Measure
3.1.2 The Class of Lebesgue Measurable Sets and the Lebesgue Measure
3.1.3 Approximating Measurable Sets
3.1.4 The Lebesgue Inner Measure
The Definition and Some Properties of the Lebesgue Inner Measure
3.1.5 The Cantor Ternary Set
The Classical CantorTernary Set
A Cantor Ternary Set of Positive Measure
3.1.6 A Nonmeasurable Set
3.1.7 Sequences of Sets

Chapter 4 Functions
4.1 Functions on Real Numbers
4.1.1 Introduction
4.1.2 The Limit of a Function
The Cauchy Criterion for the Existence of the Limit of a Function
4.1.3 Continuous Functions
4.1.4 Differentiable Functions
4.2 Optimization and the Mean Value Theorem
4.3 Algebra of Derivatives
4.4 The Trigonometric Functions
4.5 Finer Analysis of Continuity and Differentiability
4.5.1 Differentiability of the Inverse Mapping
4.5.2 Inverse Goniometric Functions
4.5.3 Monotone Functions
4.5.4 Measurable Functions
4.5.5 Differentiability of Monotone Functions
4.5.6 Functions of Bounded Variation
4.5.7 Absolutely Continuous Functions and Lipschitz Functions
4.5.8 Examples
4.5.9 The Intermediate Value Property II

Chapter 5 Function Convergence
5.1 Function Sequences
5.1.1 Pointwise and Almost Everywhere Convergence
5.1.2 Uniform Convergence
5.1.3 Convergence in Measure
5.1.4 Local Approximation by Polynomials
5.2 Function Series
5.2.1 Power Series
5.2.2 The Taylor Series
5.2.3 The Exponential and the Logarithmic Functions
Applications
5.2.4 The Hyperbolic Functions
5.2.5 The Trigonometric Functions
5.2.6 The Binomial Series

Chapter 6 Metric Spaces
6.1 Basics
6.2 Mappings Between Metric Spaces
6.3 More Examples (Continued)
6.4 Tietze's Extension Theorem
6.5 Complete Metric Spaces and the Completion of a Metric Space
6.6 Separable Metric Spaces
6.7 Polish Spaces
6.8 Compactness in Metric Spaces
6.8.1 Compact Metric Spaces
6.8.2 Total Boundedness
6.8.3 Continuous Mappings on Compact Spaces
6.8.4 The Lebesgue Number of a Covering
6.8.5 The Finite Intersection Property. Pseudocompactness
6.9 The Baire Category Theorem Continued
6.9.1 The Baire Category Theorem in the Context of Metric Spaces
6.9.2 Some Applications of the Baire Category Theorem
6.10 The Arzelà--Ascoli Theorem
6.11 Metric Fixed Point Theory
6.11.1 The Banach Contraction Principle
6.11.2 Continuity of the Fixed Point

Chapter 7 Integration
7.1 The Riemann Integral
7.1.1 Introduction
7.1.2 The Definition of the Riemann Integral
7.1.3 Properties of the Integral
7.1.4 Functions Defined by Integrals
7.1.5 Some Applications of the Riemann Integral and the Arzelà--Ascoli Theorem to the Theory of Ordinary Differential Equations
7.1.6 Some Applications of the Riemann Integral and the Fixed Point Theory to the Theory of Ordinary Differential and Integral Equations
7.1.7 Mean Value Theorems for the Riemann Integral
7.1.8 Convergence Theorems for Riemann Integrable Functions
7.1.9 Change of Variable; Integration by Parts
7.2 Improper Riemann Integrals
7.3 The Lebesgue Integral
7.3.1 Introduction
7.3.2 Step Functions
7.3.3 Upper Functions
7.3.4 Lebesgue Integrable Functions
7.3.5 Convergence Theorems
The Monotone Convergence Theorem
The Dominated Convergence Theorem
7.3.6 Measure and Integration
Measure and the Integral of a Characteristic Function
The Integral of a Function on an Arbitrary Measurable Set
The Measure Defined by a Measurable Function
7.3.7 Functions Defined by Integrals
7.3.8 The Space L1
7.3.9 Riemann versus Lebesgue Integrability, and the Riemann--Lebesgue Criterion for Riemann Integrability
The Riemann--Lebesgue Criterion for Riemann Integrability
Some Consequences of the Riemann--Lebesgue Criterion, and Some Examples
Lebesgue's Theory (Properly) Extends Riemann's Theory
Improper Riemann and Lebesgue Integrability
7.3.10 The Fundamental Theorem of Calculus for Lebesgue Integration
Introduction
The Main Result
7.3.11 Integration by Parts
7.3.12 Parametric Lebesgue Integrals

Chapter 8 Convex Functions
8.1 Basics on Convex Functions
8.2 Some Fundamental Inequalities
8.2.1 Jensen's Inequality
8.2.2 Using the Exponential Function
8.2.3 Using Powers of x (Minkowski's and Hölder's Inequalities)

Chapter 9 Fourier Series
9.1 Introduction
9.2 Some Elementary Trigonometric Identities
9.3 The Fourier Series of 2-periodic Lebesgue Integrable Functions
9.4 The Riemann--Lebesgue Lemma
9.5 The Partial Sums of a Fourier Series and the Dirichlet Kernel
9.6 Convergence of the Fourier Series
9.6.1 Pointwise Convergence of the Fourier Series
9.6.2 Cesàro Convergence of the Fourier Series
9.6.3 Uniform Convergence of the Fourier Series
9.6.4 Convergence of the Fourier Series in "026B30D "026B30D 1
9.6.5 Mean Square Convergence of the Fourier Series
9.7 The Fourier Integral

Chapter 10 Basics on Descriptive Statistics
10.1 Discrete Probability
10.1.1 Introduction
10.1.2 Random Variables
Independent Random Variables
Centrality and Dispersion of a Random Variable
Some Properties of the Mean and the Variance
10.1.3 Products of Discrete Probability Spaces
10.1.4 Inequalities
10.2 Distribution Functions
10.2.1 Selected Distributions of Discrete Random Variables
10.2.2 Continuous Random Variables and Their Distribution Functions

Chapter 11 Excursion to Functional Analysis
11.1 Real Banach Spaces
11.1.1 Spaces with a Norm (Normed Spaces, Banach Spaces)
11.1.2 Operators I
11.1.3 Finite-Dimensional Banach Spaces
11.1.4 Infinite-Dimensional Banach Spaces
Some Remarks on Compactness in Banach Spaces
11.1.5 Operators II
11.1.6 Finite-Rank and Compact Operators
11.1.7 Sets of Operators
11.2 Three Basic Principles of Linear Analysis
11.2.1 Extending Continuous Linear Functionals
The Hahn--Banach Theorem
Some Consequences of the Hahn--Banach Theorem
11.2.2 Bounded Sets of Operators
11.2.3 Continuity of the Inverse Operator
11.3 Complex Banach Spaces
11.3.1 The Associated Real Normed Space
11.3.2 Operators
11.3.3 Linear Functionals
11.3.4 Supporting Functionals and Differentiability
11.3.5 Basic Results in the Complex Setting
11.4 Spaces with an Inner Product (Pre-Hilbertian and Hilbert Spaces)
11.4.1 Basic Hilbert Space Theory
11.4.2 An Application to the Uniform Convergence of the Fourier Series
11.4.3 Complements to Hilbert Spaces
11.5 Spectral Theory
11.6 Pointwise Topology and Product Spaces
11.7 Excursion to Nonlinear Functional Analysis
11.7.1 Variational Principles
11.7.2 More on Differentiability of Convex and Lipschitz Functions
Convex Functions
Fréchet Differentiability of Convex and Lipschitz Functions
Rough Norms
11.7.3 More on Fixed Point Theorems
11.8 An Application: Periodic Distributions
11.8.1 Introduction
11.8.2 The Basic Idea
11.8.3 The Basic Definitions
The Test Functions
The Space of Periodic Distributions
11.8.4 Derivatives of Periodic Distributions
11.8.5 Convergence in PD
11.8.6 Fourier Analysis
An Application: The Heat Equation
11.9 Concluding Remarks to Chapter 11

Chapter 12 Appendix
12.1 The Set of Natural Numbers
12.2 Integer Numbers
12.3 Rational Numbers
12.4 Real Numbers
12.4.1 The Constructive Approach
12.4.2 The Axiomatic Approach
12.5 The Complex Number System
12.6 Ordering and Choice. Three Fundamental Principles in Set Theory
12.6.1 Definitions
12.6.2 Examples
12.6.3 Three Basic Principles

Chapter 13 Exercises
13.1 Numbers
13.1.1 Set-Theoretical Notations
13.1.2 Natural Numbers
13.1.3 Fractions
13.1.4 Base Representation
13.1.5 Real Numbers
13.1.6 Cardinality of Sets---and Ordinal Numbers
13.1.7 Topology of R
13.2 Sequences and Series
13.2.1 Approximation by Rational Numbers
13.2.2 Sequences
Arithmetic and Geometric Progressions
13.2.3 Series
Series of Nonnegative Terms
Series of Arbitrary Terms
13.2.4 The Euler Number e
13.3 Measure
13.3.1 The Lebesgue Outer Measure
13.3.2 The Class of Lebesgue Measurable Sets and the Lebesgue Measure
13.3.3 The Cantor Ternary Set
13.3.4 A Nonmeasurable Set
13.3.5 Sequences of Sets
13.4 Functions
13.4.1 Functions on Real Numbers
Introduction
The Limit of a Function
Continuous Functions
Uniform Continuity
The Intermediate Value Property
Differentiable Functions
13.4.2 Optimization and the Mean Value Theorem
13.4.3 The Trigonometric Functions
13.4.4 Finer Analysis of Continuity and Differentiability
Monotone Functions
Differentiability of Monotone Functions
Functions of Bounded Variation
Absolutely Continuous Functions. Lipschitz Functions
13.4.5 Function Convergence
Pointwise and Almost Everywhere Convergence
Uniform Convergence
Local Approximation by Polynomials
13.4.6 Function Series
Power Series
The Taylor Series
The Exponential and the Logarithmic Functions
The Hyperbolic Functions
The Trigonometric Functions
13.4.7 Metric Spaces
Basics
Mappings Between Metric Spaces
Complete Metric Spaces, and the Completion of a Metric Space
Separable Metric Spaces
Polish Spaces
Compactness in Metric Spaces
The Baire Category Theorem
The Arzelà--Ascoli Theorem
Metric Fixed Point Theory
13.5 Integration
13.5.1 The Riemann Integral
Basics
Functions Defined by Integrals
Convergence Theorems
Change of Variable. Integration by Parts
Parametric Integrals
13.5.2 Review of Some Frequently used Techniques for calculating Antiderivatives
Integration by Parts
Rational Functions
Algebraic Irrationals
Binomial Integrals
Integration of Transcendental Functions
An Alternative Way to Roots
A Special Case of Trigonometric Polynomials
13.5.3 Improper Riemann Integral
13.5.4 Notes on Vector-Valued Riemann Integration
13.5.5 The Lebesgue Integral
Basics
Convergence Theorems
Measure and Integration
Functions Defined by Integrals
Riemann Versus Lebesgue Integrability
The Fundamental Theorem of Calculus for the Lebesgue Integral
13.5.6 Convex Functions
13.6 Fourier Series
13.7 Basics on Descriptive Statistics
13.8 Excursion to Functional Analysis
13.8.1 Banach Spaces
Basics
Equivalent Norms
13.8.2 Operators
Linear Functionals
Operators
13.8.3 Finite-Dimensional Spaces
13.8.4 Infinite-Dimensional Spaces
13.8.5 Operators II
Finite-Rank and Compact Operators
Sets of Operators
13.8.6 Three Principles of Linear Analysis
Dual Spaces
Extreme Points and Exposed Points in Banach Spaces
Differentiability
The Banach--Steinhaus Theorem
The Open Mapping and Closed Graph Theorems
13.8.7 Spaces with an Inner Product (Pre-Hilbertian and Hilbert spaces)
Basics
Extreme Points and Exposed in Hilbert Spaces
Differentiability in Hilbert Spaces
13.8.8 Spectral Theory
13.8.9 Pointwise Topology and Product Spaces
13.8.10 Periodic Distributions

References
Author Index
General Index
Symbol Index


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