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  • Textbook
  • © 2016

Analysis I

Third Edition

Authors:

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  • Discusses all major topics of analysis in a simple, lucid manner

  • Highlights the concrete setting of the real line and Euclidean spaces

  • Provides examples, and step-by-step instructions

  • Evolves from the author’s lectures delivered at the University of California, Los Angeles

  • Covers about 30 lectures, this is part one of the two-volume book

  • Is authored by the winner of the Fields Medal

  • Includes supplementary material: sn.pub/extras

Part of the book series: Texts and Readings in Mathematics (TRIM, volume 37)

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USD 59.99
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  • ISBN: 978-981-10-1789-6
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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xix
  2. Introduction

    • Terence Tao
    Pages 1-12
  3. Set theory

    • Terence Tao
    Pages 33-73
  4. Integers and Rationals

    • Terence Tao
    Pages 74-93
  5. The real numbers

    • Terence Tao
    Pages 94-125
  6. Limits of sequences

    • Terence Tao
    Pages 126-154
  7. Series

    • Terence Tao
    Pages 155-180
  8. Infinite sets

    • Terence Tao
    Pages 181-210
  9. Continuous functions on R

    • Terence Tao
    Pages 211-250
  10. Differentiation of functions

    • Terence Tao
    Pages 251-266
  11. The Riemann integral

    • Terence Tao
    Pages 267-304
  12. Back Matter

    Pages 305-350

About this book

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.        

Keywords

  • natural numbers
  • Set theory
  • Integers and rationals
  • real numbers
  • Limits of sequences
  • Series
  • Continuous functions
  • Dierentiation of functions
  • Riemann integral

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Authors and Affiliations

  • Department of Mathematics, UCLA, Los Angeles, USA

    Terence Tao

About the author

Terence "Terry" Chi-Shen Tao, FAA FRS, is an Australian mathematician. His areas of interests are in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Professor Tao is a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. He maintains a personal mathematics blog, which has been described by Timothy Gowers as “the undisputed king of all mathematics blogs”.  

Bibliographic Information

Buying options

eBook
USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-981-10-1789-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout