Real analysis
Author(s)
Bibliographic Information
Real analysis
Birkhäuser , Springer, c2016
Available at 5 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
  Kanagawa
  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
  Shiga
  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
  Kagoshima
  Okinawa
  Korea
  China
  Thailand
  United Kingdom
  Germany
  Switzerland
  France
  Belgium
  Netherlands
  Sweden
  Norway
  United States of America
Note
Includes bibliographical references (p. 267-269) and index
Description and Table of Contents
Description
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.
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.
by "Nielsen BookData"