Real Analysis

Nonfiction, Science & Nature, Mathematics, Functional Analysis, Mathematical Analysis
Cover of the book Real Analysis by Peter A. Loeb, Springer International Publishing
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Peter A. Loeb ISBN: 9783319307442
Publisher: Springer International Publishing Publication: May 5, 2016
Imprint: Birkhäuser Language: English
Author: Peter A. Loeb
ISBN: 9783319307442
Publisher: Springer International Publishing
Publication: May 5, 2016
Imprint: Birkhäuser
Language: English

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. 

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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. 

More books from Springer International Publishing

Cover of the book Friedrich List’s Exile in the United States by Peter A. Loeb
Cover of the book Managing Ultrasonography in Human Reproduction by Peter A. Loeb
Cover of the book Device Physics, Modeling, Technology, and Analysis for Silicon MESFET by Peter A. Loeb
Cover of the book Rapidly Changing Securities Markets by Peter A. Loeb
Cover of the book Climate Change, Glacier Response, and Vegetation Dynamics in the Himalaya by Peter A. Loeb
Cover of the book Infrared Thermography for Building Moisture Inspection by Peter A. Loeb
Cover of the book Compressive Sensing Based Algorithms for Electronic Defence by Peter A. Loeb
Cover of the book Multi-Disciplinary Engineering for Cyber-Physical Production Systems by Peter A. Loeb
Cover of the book Perioperative Medical Management for Total Joint Arthroplasty by Peter A. Loeb
Cover of the book Artificial Intelligence and Soft Computing by Peter A. Loeb
Cover of the book Central Banking and Financial Stability in East Asia by Peter A. Loeb
Cover of the book Theatricality and Performativity by Peter A. Loeb
Cover of the book Linguistic Disobedience by Peter A. Loeb
Cover of the book Set-Theoretic Methods in Control by Peter A. Loeb
Cover of the book Principles of Osteoimmunology by Peter A. Loeb
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy