UID:
almahu_9949698031602882
Format:
1 online resource (247 p.)
ISBN:
1-281-76367-5
,
9786611763671
,
0-08-087437-1
Series Statement:
Pure and Applied Mathematics
Content:
The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstan
Note:
Description based upon print version of record.
,
Front Cover; An Introduction to Nonstandard Real Analysis; Copyright Page; Contents; Preface; Chapter I. lnfinitesimals and The Calculus; I.1 The Hyperreal Number System as an Ultrapower; I.2 *-Transforms of Relations; I.3 Simple Languages for Relational Systems; I.4 Interpretation of Simple Sentences; I.5 The Transfer Principle for Simple Sentences; I.6 Infinite Numbers, Infinitesimals, and the Standard Part Map; I.7 The Hyperintegers; I.8 Sequences and Series; I.9 Topology on the Reals; I.10 Limits and Continuity; I.11 Differentiation; I.12 Riemann Integration; I.13 Sequences of Functions
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I.14 Two Applications to Differential EquationsI.15 Proof of the Transfer Principle; Chapter II. Nonstandard Analysis on Superstructures; II.1 Superstructures; II.2 Languages and Interpretation for Superstructures; II.3 Monomorphisms between Superstructures: The Transfer Principle; II.4 The Ultrapower Construction for Superstructures; II.5 Hyperfinite Sets, Enlargements, and Concurrent Relations; II.6 Internal and External Entities; Comprehensiveness; II.7 The Permanence Principle; II.8 ?-Saturated Superstructures; Chapter III. Nonstandard Theory of Topological Spaces
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III.1 Basic Definitions and ResultsIII.2 Compactness; III.3 Metric Spaces; III.4 Normed Vector Spaces and Banach Spaces; III.5 Inner-Product Spaces and Hibert Spaces; III.6 Nonstandard Hulls of Metric Spaces; III.7 Compactifications; III.8 Function Spaces; Chapter IV. Nonstandard Integration Theory; IV.1 Standardizations of Internal Integration Structures; IV.2 Measure Theory for Complete Integration Structures; IV.3 Integration on R""; the Riesz Representation Theorem; IV.4 Basic Convergence Theorems; IV.5 The Fubini Theorem; IV.6 Applications to Stochastic Processes; Appendix: Ultrafilters
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ReferencesList of Symbols; Index; Pure and Applied Mathematics
,
English
Additional Edition:
ISBN 0-12-362440-1
Language:
English
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