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

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## Eric Schechter:

## Eric Schechter: *Handbook of Analysis and its Foundations*

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

- ISBN 0126227608.

### Subject Matter

### 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**