Book:Eric Schechter/Handbook of Analysis and its Foundations

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