Principles of real analysis: measure, integration, functional analysis, and applications

Content: Cover --
Half Title --
Title --
Copyright --
Dedication --
Contents --
Preface --
0 Preliminaries --
0.1 Sets --
Set Operations --
Number Systems --
Relations --
Functions --
Cardinality --
0.2 Algebraic Structures --
Semigroups and Groups --
Linear Spaces --
Linear Transformations --
Quotient Linear Spaces --
Algebras --
0.3 Metric Spaces --
Open and Closed Sets --
Interior, Closure, and Boundary --
Sequential Convergence. Completeness --
Continuity --
Category --
0.4 Normed Linear Spaces --
Norms and Seminorms --
Banach Spaces --
Completion of a Normed Space --
In nite Series in Normed Spaces --
Unordered Sums in Normed Spaces --
Bounded Linear Transformations --
Banach Algebras --
0.5 Topological Spaces --
Open and Closed Sets --
Neighborhood Systems --
Neighborhood Bases --
Relative Topology --
Nets --
0.6 Continuity in Topological Spaces --
De nition and General Properties --
Initial Topologies --
Product Topology --
Final Topologies --
Quotient Topology --
The Space of Continuous Functions --
F-sigma and G-delta Sets --
0.7 Normal Topological Spaces --
Urysohn's Lemma --
Tietze Extension Theorem --
0.8 Compact Topological Spaces --
Convergence in Compact Spaces --
Compactness of Cartesian Products --
Continuity and Compactness --
0.9 Totally Bounded Metric Spaces --
0.10 Equicontinuity --
0.11 The Stone-Weierstrass Theorem --
0.12 Locally Compact Topological Spaces --
General Properties --
Functions with Compact Support --
Functions That Vanish at In nity --
The One-Point Compacti cation --
0.13 Spaces of Di erentiable Functions --
0.14 Partitions of Unity --
0.15 Connectedness --
I Measure and Integration --
1 Measurable Sets --
1.1 Introduction --
1.2 Measurable Spaces --
Fields and Sigma Fields --
Generated Sigma Fields --
Borel Sets --
Extended Borel Sets --
Product Sigma Fields --
Pi-Systems and Lambda-Systems --
Exercises. 1.3 Measures --
Set Functions --
Properties and Examples of Measures --
Exercises --
1.4 Complete Measure Spaces --
Completion Theorem --
Null Sets --
Exercises --
1.5 Outer Measure and Measurability --
Construction of an Outer Measure --
Carath eodory's Theorem --
Exercises --
1.6 Extension of a Measure --
The Measure Extension Theorem --
Approximation Property of the Extension --
Completeness of the Extension --
Uniqueness of the Extension --
Exercises --
1.7 Lebesgue Measure --
The Volume Set Function --
Construction of the Measure --
Exercises --
1.8 Lebesgue-Stieltjes Measures --
Regularity --
One-Dimensional Distribution Functions --
Higher Dimensional Distribution Functions --
Exercises --
*1.9 Some Special Sets --
An Uncountable Set with Lebesgue Measure Zero --
Non-Lebesgue-Measurable Sets --
A Lebesgue Measurable, Non-Borel Set --
Exercises --
2 Measurable Functions --
2.1 Measurable Transformations --
General Properties --
Exercises --
2.2 Measurable Numerical Functions --
Criteria for Measurability --
Almost Everywhere Properties --
Combinatorial and Limit Properties of Measurable Functions --
Exercises --
2.3 Simple Functions --
A Fundamental Convergence Theorem --
Applications --
Exercises --
2.4 Convergence of Measurable Functions --
Modes of Convergence --
Relationships Among the Modes of Convergence --
Exercises --
3 Integration --
3.1 Construction of the Integral --
Integral of a Nonnegative Simple Function --
Integral of a Real-Valued Function --
Integral of a Complex-Valued Function --
Integral over a Measurable Set --
3.2 Basic Properties of the Integral --
Almost Everywhere Properties --
Monotone Convergence Theorem --
Linearity of the Integral --
Integration Against an Image Measure --
Integration Against a Measure with Density --
Change of Variables Theorem --
Exercises --
3.3 Connections with the Riemann Integral on Rd. The Darboux Integral --
The Riemann Integral --
Measure Zero Criterion for Riemann Integrability --
Improper Riemann Integrals --
Exercises --
3.4 Convergence Theorems --
The General Monotone Convergence Theorem --
Fatou's Lemma --
The Dominated Convergence Theorem --
Exercises --
3.5 Integration against a Product Measure --
Construction of the Product of Two Measures --
Fubini's Theorem --
The d-Dimensional Case --
Exercises --
3.6 Applications of Fubini's Theorem --
Gaussian Density --
Integration by Parts --
Spherical Coordinates --
Volume of a d-Dimensional Ball --
Integration of Radial Functions --
Surface Area of a d-Dimensional Ball --
Exercises --
4 Lp Spaces --
4.1 De nition and General Properties --
The Case 1 p <
1 --
The Case p = 1 --
The Case 0 <
p <
1 --
`p-Spaces --
Exercises --
4.2 Lp Approximation --
Approximation by Simple Functions --
Approximation by Continuous Functions --
Approximation by Step Functions --
Exercises --
4.3 Lp Convergence --
Exercises --
*4.4 Uniform Integrability --
Exercises --
*4.5 Convex Functions and Jensen's Inequality --
Exercises --
5 Di erentiation --
5.1 Signed Measures --
De nition and a Fundamental Example --
The Hahn-Jordan Decomposition --
Exercises --
5.2 Complex Measures --
The Total Variation Measure --
The Vitali-Hahn-Saks Theorem --
The Banach Space of Complex Measures --
Integration against a Signed or Complex Measure --
Exercises --
5.3 Absolute Continuity of Measures --
General Properties of Absolute Continuity --
The Radon-Nikodym Theorem --
Lebesgue-Decomposition of a Measure --
Exercises --
5.4 Di erentiation of Measures --
De nition and Properties of the Derivative --
Connections with the Classical Derivative --
Existence of the Measure Derivative --
Exercises --
5.5 Functions of Bounded Variation --
De nition and Basic Properties --
The Total Variation Function. Di erentiation of Functions of Bounded Variation --
Exercises --
5.6 Absolutely Continuous Functions --
De nition and Basic Properties --
Fundamental Theorems of Calculus --
Exercises --
6 Fourier Analysis on Rd --
6.1 Convolution of Functions --
De nition and Basic Properties --
Approximate Identities --
Exercises --
6.2 The Fourier Transform --
De nition and Basic Properties --
The Fourier Inversion Theorem --
Exercises --
6.3 Rapidly Decreasing Functions --
De nition and Basic Properties --
The Plancherel Theorem --
Exercises --
6.4 Fourier Analysis of Measures on Rd --
Convolution of Measures --
The Fourier-Stieltjes Transform --
Exercises --
7 Measures on Locally Compact Spaces --
7.1 Radon Measures --
De nition and Basic Properties --
Consequences of Regularity --
The Space of Complex Radon Measures --
The Support of a Radon Measure --
Exercises --
7.2 The Riesz Representation Theorem --
Exercises --
7.3 Products of Radon Measures --
Finitely Many Measures --
In nitely Many Measures --
Exercises --
7.4 Vague Convergence --
Exercises --
*7.5 The Daniell-Stone Representation Theorem --
II Functional Analysis --
II Functional Analysis 197 --
8 Banach Spaces --
8.1 General Properties of Normed Spaces --
Topology and Geometry --
Separable Spaces --
Equivalent Norms --
Finite Dimensional Spaces --
Strictly Convex Spaces --
Exercises --
8.2 Bounded Linear Transformations --
The Operator Norm --
The Banach Algebra B(X --
The Dual Space X0 --
Bilinear Transformations --
Exercises --
8.3 Concrete Representations of Dual Spaces --
The Dual of c0 --
The Dual of c --
The Dual of Lp --
The Dual of C0(X --
Exercises --
8.4 Some Constructions --
Product Spaces --
Direct Sums --
Quotient Spaces --
Exercises --
8.5 Hahn-Banach Extension Theorems --
Real Version --
Complex Version --
Normed Space Version --
The Bidual of a Normed Space --
Invariant Versions. Exercises --
*8.6 Applications of the Hahn-Banach Theorem --
The Moment Problem --
Invariant Means --
Banach Limits --
Invariant Set Functions --
Exercises --
8.7 Baire Category in Banach Spaces --
The Uniform Boundedness Principle --
The Open Mapping Theorem --
The Closed Graph Theorem --
Exercises --
*8.8 Applications --
Divergent Fourier Series --
Vector-Valued Analytic Functions --
Summability --
Schauder Bases --
Exercises --
8.9 The Dual Operator --
De nition and Properties --
Annihilators --
Duals of Quotient Spaces and Subspaces --
Exercises --
8.10 Compact Operators --
Fredholm Alternative for Compact Operators --
Exercises --
9 Locally Convex Spaces --
9.1 General Properties --
Geometry and Topology --
Seminormed Spaces --
Fr echet Spaces --
Exercises --
9.2 Continuous Linear Functionals --
Continuity on Topological Vector Spaces --
Continuity on Locally Convex Spaces --
Continuity on Finite Dimensional Spaces --
Exercises --
9.3 Hahn-Banach Separation Theorems --
Weak Separation in a TVS --
Strict Separation in a LCS --
Some Consequences of the Separation Theorems --
The Bipolar Theorem --
Exercises --
*9.4 Some Constructions --
Product Spaces --
Quotient Spaces --
Strict Inductive Limits --
Exercises --
10 Weak Topologies on Normed Spaces --
10.1 The Weak Topology --
De nition and General Properties --
Weak Sequential Convergence --
Convexity and Closure --
Application: Weak Bases --
Exercises --
10.2 The Weak Topology --
De nition and General Properties --
The Dual of X0 w --
The Banach-Alaoglu Theorem --
Application: Means on Function Spaces --
Weak Continuity --
The Closed Range Theorem --
Exercises --
10.3 Re exive Spaces --
Examples and Basic Properties --
Weak Compactness and Re exivity --
Exercises --
*10.4 Uniformly Convex Spaces --
De nition and General Properties --
Connections with Strict Convexity.