# Book:Eric Schechter/Handbook of Analysis and its Foundations

## Eric Schechter: Handbook of Analysis and its Foundations

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

ISBN 0126227608

### Contents

Preface
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
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
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
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
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
Continuity
Neighborhood Bases and Topology Bases
Cluster Points
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
Integrals of Continuous Functions
Monotone Convergence Theorem
Absolute Integrability
Henstock and Lebesgue Integrals
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
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
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