In the early 1960s, by using techniques from the model theory of first-order logic, Robinson gave a rigorous formulation and extension of Leibniz' infinitesimal calculus. Since then, the methodology has found applications in a wide spectrum of areas in mathematics, with particular success in the probability theory and functional analysis. In the latter, fruitful results were produced with Luxemburg's invention of the nonstandard hull construction. However, there is still no publication of a coherent and self-contained treatment of functional analysis using methods from nonstandard analysis. This publication aims to fill this gap.
Readership: Graduate level students and researchers in functional analysis.
Author(s): Siu-ah Ng
Publisher: World Scientific Publishing Company
Year: 2010
Language: English
Pages: C, XXII, 316
Cover
S Title
NONSTANDARD METHODS IN FUNCTIONAL ANALYSIS, lectures and notes
Copyright
© 2010 by World Scientific Publishing Co. Pte. Ltd.
ISBN-13 978-981-4287-54-8
ISBN-10 981-4287-54-7
QA321 .N47 2010 515'.7--dc22
LCCN 2010004689
Dedication
Preface
Convention and Symbols
List of Theorems
Contents
Chapter 1 Nonstandard Analysis
1.1 Sets and Logic
1.1.1 Na ve sets, rst order formulas and ZFC
1.1.2 First order theory and consistency
1.1.3 In nities, ordinals, cardinals and AC
1.1.4 Notes and exercises
Exercises
1.2 The Nonstandard Universe
1.2.1 Elementary extensions and saturation
1.2.1 Elementary extensions and saturation
1.2.2 Superstructure, internal and external sets
1.2.3 Two principles
1.2.4 Internal extensions
1.2.5 Notes and exercises
Exercises
1.3 The Ultraproduct Construction
1.3.1 Notes and exercises
Exercises
1.4 Application: Elementary Calculus
1.4.1 In nite, in nitesimals and the standard part
1.4.2 Overspill, underspill and limits
1.4.3 In nitesimals and continuity
1.4.4 Notes and exercises
Exercises
1.5 Application: Measure Theory
1.5.1 Classical measures
1.5.2 Internal measures and Loeb measures
1.5.3 Lebesgue measure, probability and liftings
1.5.4 Measure algebras and Kelley's Theorem
1.5.5 Notes and exercises
Exercises
1.6 Application: Topology
1.6.1 Monads and topologies
1.6.2 Monads and separation axioms
1.6.3 Standard part and continuity
1.6.4 Robinson's characterization of compactness
1.6.5 The Baire Category Theorem
1.6.6 Stone- Cech compacti cation
1.6.7 Notes and exercises
Exercises
Chapter 2 Banach Spaces
2.1 Norms and Nonstandard Hulls
2.1.1 Seminormed linear spaces and quotients
2.1.2 Internal spaces and nonstandard hulls
2.1.3 Finite dimensional Banach spaces
2.1.4 Examples of Banach spaces
2.1.5 Notes and exercises
Exercises
2.2 Linear Operators and Open Mappings
2.2.1 Bounded linear operators and dual spaces
2.2.2 Open mappings
2.2.3 Uniform boundedness
2.2.4 Notes and exercises
Exercises
2.3 Helly's Theorem and the Hahn-Banach Theorem
2.3.1 Norming and Helly's Theorem
2.3.2 The Hahn-Banach Theorem
2.3.3 The Hahn-Banach Separation Theorem
2.3.4 Notes and exercises
Exercises
2.4 General Nonstandard Hulls and Biduals
2.4.1 Nonstandard hulls by internal seminorms
2.4.2 Weak nonstandard hulls and biduals
2.4.3 Applications of weak nonstandard hulls
2.4.4 Weak compactness and separation
2.4.5 Weak* topology and Alaoglu's Theorem
2.4.6 Notes and exercises
Exercises
2.5 Reexive Spaces
2.5.1 Weak compactness and reexivity
2.5.2 The Eberlein- Smulian Theorem
2.5.3 James' characterization of reexivity
2.5.4 Finite representability and superreexivity
2.5.5 Notes and exercises
Exercises
2.6 Hilbert Spaces
2.6.1 Basic properties
2.6.2 Examples
2.6.3 Notes and exercises
Exercises
2.7 Miscellaneous Topics
2.7.1 Compact operators
2.7.2 The Krein-Milman Theorem
2.7.3 Schauder bases
2.7.4 Schauder's Fixed Point Theorem
2.7.5 Notes and exercises
Exercises
Chapter 3 Banach Algebras
3.1 Normed Algebras and Nonstandard Hulls
3.1.1 Examples and basic properties
3.1.2 Spectra
3.1.3 Nonstandard hulls
3.1.4 Notes and exercises
Exercises
3.2 C*-Algebras
3.2.1 Examples and basic properties
3.2.2 The Gelfand transform
3.2.3 The GNS construction
3.2.4 Notes and exercises
Exercises
3.3 The Nonstandard Hull of a C*-Algebra
3.3.1 Basic properties
3.3.2 Notes and exercises
Exercises
3.4 Von Neumann Algebras
3.4.1 Operator topologies and the bicommutant
3.4.2 Nonstandard hulls vs. von Neumann algebras
3.4.3 Weak nonstandard hulls and biduals
3.4.4 Notes and exercises
Exercises
3.5 Some Applications of Projections
3.5.1 Infinite C*-algebras
3.5.2 P*-algebras
3.5.3 Notes and exercises
Exercises
Chapter 4 Selected Research Topics
4.1 Hilbert space-valued integrals
4.2 Reflexivity and fixed points
4.3 Arens product on a bidual
4.4 Noncommutative Loeb measures
4.5 Further questions and problems
Suggestions for Further Reading
Bibliography
Index
