Book:Eric Schechter/Handbook of Analysis and its Foundations

Eric Schechter: Handbook of Analysis and its Foundations

Published $\text {1996}$, Elsevier Inc.

ISBN 0126227608

Subject Matter

• Analysis

Contents

Preface

About the Choice of Topics

Existence, Examples, and Intangibles

Abstract versus Concrete

Order of Topics

How to Use This Book

Acknowledgements

To Contact Me

A SETS AND ORDERINGS

1 Sets

Mathematical Language and Informal Logic

Basic Notations for Sets

Ways to Combine Sets

Functions and Products of Sets

ZF Set Theory

2 Functions

Some Special Functions

Distances

Cardinality

Induction and Recursion on the Integers

3 Relations and Orderings

Relations

Preordered Sets

More about Equivalences

More about Posets

Max, Sup, and Other Special Elements

Chains

Van Maaren's Geometry-Free Sperner Lemma

Well Ordered Sets

4 More about Sups and Infs

Moore Collections and Moore Closures

Some Special Types of Moore Closures

Lattices and Completeness

More about Lattices

More about Complete Lattices

Order Completions

Sups and Infs in Metric Spaces

5 Filters, Topologies, and Other Sets of Sets

Filters and Ideals

Topologies

Algebras and Sigma-Algebras

Uniformities

Images and Preimages of Sets of Sets

Transitive Sets and Ordinals

The Class of Ordinals

6 Constructivism and Choice

Examples of Nonconstructive Mathematics

Further Comments on Constructivism

The Meaning of Choice

Variants and Consequences of Choice

Some Equivalents of Choice

Countable Choice

Dependent Choice

The Ultrafilter Principle

7 Nets and Convergences

Nets

Subnets

Universal Nets

More about Subsequences

Convergence Spaces

Convergence in Posets

Convergence in Complete Lattices

B ALGEBRA

8 Elementary Algebraic Systems

Monoids

Groups

Sums and Quotients of Groups

Rings and Fields

Matrices

Ordered Groups

Lattice Groups

Universal Algebras

Examples of Equational Varieties

9 Concrete Categories

Definitions and Axioms

Examples of Categories

Initial Structures and Other Categorical Constructions

Varieties with Ideals

Functors

The Reduced Power Functor

Exponential (Dual) Functors

10 The Real Numbers

Dedekind Completions of Ordered Groups

Ordered Fields and the Reals

The Hyperreal Numbers

Quadratic Extensions and the Complex Numbers

Absolute Values

Convergence of Sequences and Series

11 Linearity

Linear Spaces and Linear Subspaces

Linear Maps

Linear Dependence

Further Results in Finite Dimensions

Choice and Vector Bases

Dimension of the Linear Dual (Optional)

Preview of Measure and Integration

Ordered Vector Spaces

Positive Operators

Orthogonality in Riesz Spaces (Optional)

12 Convexity

Convex Sets

Combinatorial Convexity in Finite Dimensions (Optional)

Convex Functions

Norms, Balanced Functionals, and Other Special Functions

Minkowski Functionals

Hahn-Banach Theorems

Convex Operators

13 Boolean Algebras

Boolean Lattices

Boolean Homomorphisms and Subalgebras

Boolean Rings

Boolean Equivalents of UF

Heyting Algebras

14 Logic and Intangibles

Some Informal Examples of Models

Languages and Truths

Ingredients of First-Order Language

Assumptions in First-Order Logic

Some Syntactic Results (Propositional Logic)

Some Syntactic Results (Predicate Logic)

The Semantic View

Soundness, Completeness, and Compactness

Nonstandard Analysis

Summary of Some Consistency Results

Quasiconstructivism and Intangibles

C TOPOLOGY AND UNIFORMITY

15 Topological Spaces

Pretopological Spaces

Topological Spaces and Their Convergences

More about Topological Spaces

Continuity

Neighborhood Bases and Topology Bases

Cluster Points

More about Intervals

16 Separation and Regularity Axioms

Kolmogorov (T-Zero) Topologies and Quotients

Symmetric and Fréchet (T-One) Topologies

Preregular and Hausdorff (T-Two) Topologies

Regular and T-Three Topologies

Completely Regular and Tychonov (T-Three and a Half) Topologies

Partitions of Unity

Normal Topologies

Paracompactness

Hereditary and Productive Properties

17 Compactness

Characterization in Terms of Convergences

Basic Properties of Compactness

Regularity and Compactness

Tychonov's Theorem

Compactness and Choice (Optional)

Compactness, Maxima, and Sequences

Pathological Examples: Ordinal Spaces (Optional)

Boolean Spaces

Eberlein-Smulian Theorem

18 Uniform Spaces

Lipschitz Mappings

Uniform Continuity

Pseudometrizable Gauges

Compactness and Uniformity

Uniform Convergence

Equicontinuity

19 Metric and Uniform Completeness

Cauchy Filters, Nets, and Sequences

Complete Metrics and Uniformities

Total Boundedness and Precompactness

Bounded Variation

Cauchy Continuity

Cauchy Spaces (Optional)

Completions

Banach's Fixed Point Theorem

Meyers's Converse (Optional)

Bessaga's Converse and Brönsted's Principle (Optional)

20 Baire Theory

G-Delta Sets

Meager Sets

Generic Continuity Theorems

Topological Completeness

Baire Spaces and the Baire Category Theorem

Almost Open Sets

Relativization

Almost Homeomorphisms

Tail Sets

Baire Sets (Optional)

21 Positive Measure and Integration

Measurable Functions

Joint Measurability

Positive Measures and Charges

Null Sets

Lebesgue Measure

Some Countability Arguments

Convergence in Measure

Integration of Positive Functions

Essential Suprema

D TOPOLOGICAL VECTOR SPACES

22 Norms

(G-)(Semi-)Norms

Basic Examples

Sup Norms

Convergent Series

Bochner-Lebesgue Spaces

Strict Convexity and Uniform Convexity

Hilbert Spaces

23 Normed Operators

Norms of Operators

Equicontinuity and Joint Continuity

The Bochner Integral

Hahn-Banach Theorems in Normed Spaces

A Few Consequences of HB

Duality and Separability

Unconditionally Convergent Series

Neumann Series and Spectral Radius (Optional)

24 Generalized Riemann Integrals

Definitions of the Integrals

Basic Properties of Gauge Integrals

Additivity over Partitions

Integrals of Continuous Functions

Monotone Convergence Theorem

Absolute Integrability

Henstock and Lebesgue Integrals

More about Lebesgue Measure

More about Riemann Integrals (Optional)

25 Fréchet Derivatives

Definitions and Basic Properties

Partial Derivatives

Strong Derivatives

Derivatives of Integrals

Integrals of Derivatives

Some Applications of the Second Fundamental Theorem of Calculus

Path Integrals and Analytic Functions (Optional)

26 Metrization of Groups and Vector Spaces

F-Seminorms

TAG's and TVS's

Arithmetic in TAG's and TVS's

Neighborhoods of Zero

Characterizations in Terms of Gauges

Uniform Structure of TAG's

Pontryagin Duality and Haar Measure (Optional; Proofs Omitted)

Ordered Topological Vector Spaces

27 Barrels and Other Features of TVS's

Bounded Subsets of TVS's

Bounded Sets in Ordered TVS's

Dimension in TVS's

Fixed Point Theorems of Brouwer, Shauder, and Tychonov

Barrels and Ultrabarrels

Proofs of Barrel Theorems

Inductive Topologies and LF Spaces

The Dream Universe of Garnir and Wright

28 Duality and Weak Compactness

Hahn-Banach Theorems in TVS's

Bilinear Pairings

Weak Topologies

Weak Topologies of Normed Spaces

Polar Arithmetic and Equicontinuous Sets

Duals of Product Spaces

Characterizations of Weak Compactness

Some Consequences in Banach Spaces

More about Uniform Convexity

Duals of the Lebesgue Spaces

29 Vector Measures

Basic Properties

The Variation of a Charge

Indefinite Bochner Integrals and Radon-Nikodym Derivatives

Conditional Expectations and Martingales

Existence of Radon-Nikodym Derivatives

Semivariation and Bartle Integrals

Measures on Intervals

Pincus's Pathology (Optional)

30 Initial Value Problems

Elementary Pathological Examples

Carathéodory Solutions

Lipschitz Conditions

Generic Solvability

Compactness Conditions

Isotonicity Conditions

Generalized Solutions

Semigroups and Dissipative Operators

References

Index and Symbol List