Nonstandard methods in functional analysis

Nonstandard methods in functional analysis: lectures and notes

by Ng, Siu-Ah

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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 coh…

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Title Statements

  • Full title :Nonstandard methods in functional analysis : lectures and notes / Siu-Ah Ng.

Summary

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.

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Edition Statements

  • Publication Date:©2010.

Publication Statements

The Physical Object

  • Description:1 online resource (xxii, 316 pages) : : illustrations

Edition Identifiers

  • ISBN:9789814287555, 9814287555, 1282761544, 9781282761544, 9786612761546, 6612761547
  • oclc:ocn754712121
  • LC Classification:QA321
  • Dewey Decimal:515/.7

Table of Contents

1. Nonstandard analysis. 1.1. Sets and logic. 1.2. The nonstandard universe. 1.3. The ultraproduct construction. 1.4. Application : elementary calculus. 1.5. Application : measure theory. 1.6. Application : topology -- 2. Banach spaces. 2.1. Norms and nonstandard hulls. 2.2. Linear operators and open mappings. 2.3. Helly's theorem and the Hahn-Banach theorem. 2.4. General nonstandard hulls and biduals. 2.5. Reflexive spaces. 2.6. Hilbert spaces. 2.7. Miscellaneous topics -- 3. Banach algebras. 3.1. Normed algebras and nonstandard hulls. 3.2. C*-algebras. 3.3. The nonstandard hull of a C*-algebra. 3.4. Von Neumann algebras. 3.5. Some applications of projections -- 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.

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Subjects

Analyse fonctionnelle Analyse mathématique non standard Functional analysis Nonstandard mathematical analysis

Writers

Ng, Siu-Ah

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