Real analysis [electronic resource] / Peter A Loeb.
2016
QA300
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Title
Real analysis [electronic resource] / Peter A Loeb.
Author
Loeb, Peter A., author.
ISBN
9783319307442 (electronic book)
3319307444 (electronic book)
9783319307428
3319307444 (electronic book)
9783319307428
Published
Switzerland : Birkhäuser, 2016.
Language
English
Description
1 online resource (xii, 274 pages)
Item Number
10.1007/978-3-319-30744-2 doi
Call Number
QA300
Dewey Decimal Classification
515
Summary
This textbook is designed for a year-long course in real analysis taken by beginning graduate and advanced undergraduate students in mathematics and other areas such as statistics, engineering, and economics. Written by one of the leading scholars in the field, it elegantly explores the core concepts in real analysis and introduces new, accessible methods for both students and instructors. The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach. The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics. .
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed May 11, 2016).
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Table of Contents
Preface
Set Theory and Numbers
Measure on the Real Line
Measurable Functions
Integration
Differentiation and Integration
General Measure Spaces
Introduction to Metric and Normed Spaces
Hilbert Spaces
Topological Spaces
Measure Construction
Banach Spaces
Appendices
References. .
Set Theory and Numbers
Measure on the Real Line
Measurable Functions
Integration
Differentiation and Integration
General Measure Spaces
Introduction to Metric and Normed Spaces
Hilbert Spaces
Topological Spaces
Measure Construction
Banach Spaces
Appendices
References. .