Real Analysis

Peter A. Loeb

ISBN 10: 3319307428 / ISBN 13: 9783319307428

Published by Springer International Publishing, 2016

About this Item

Gebraucht - Sehr gut SG - Ungelesenes Mängelexemplar, gestempelt, mit leichten Lagerspuren - 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.

Seller Inventory # INF1000553655

Bibliographic Details

• Title: Real Analysis
• Publisher: Springer International Publishing
• Publication Date: 2016
• Language: English
• ISBN 10: 3319307428
• ISBN 13: 9783319307428
• Binding: Gebundene Ausgabe
• Condition: Sehr gut
• Dimensions: 241W x 160H x 20D millimeters 241 millimeters width by 160 millimeters height by 20 millimeters depth
• Weight: 652 grams

About this title

Synopsis

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. 

About the Author

Peter Loeb is an emeritus Professor of Mathematics at the University of Illinois in Champaign-Urbana. His research is centered on problems of real analysis and applications of model theory to real analysis.