© 2016

Analysis I

Third Edition


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

Table of contents

  1. Front Matter
    Pages i-xix
  2. Terence Tao
    Pages 1-12
  3. Terence Tao
    Pages 33-73
  4. Terence Tao
    Pages 74-93
  5. Terence Tao
    Pages 94-125
  6. Terence Tao
    Pages 126-154
  7. Terence Tao
    Pages 155-180
  8. Terence Tao
    Pages 181-210
  9. Terence Tao
    Pages 211-250
  10. Terence Tao
    Pages 251-266
  11. 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.        


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

Authors and affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

About the authors

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

  • Book Title Analysis I
  • Book Subtitle Third Edition
  • Authors Terence Tao
  • Series Title Texts and Readings in Mathematics
  • Series Abbreviated Title Texts and Readings in Mathematics
  • DOI
  • Copyright Information Springer Science+Business Media Singapore 2016
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • eBook ISBN 978-981-10-1789-6
  • Series ISSN 2366-8717
  • Series E-ISSN 2366-8725
  • Edition Number 1
  • Number of Pages XIX, 350
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
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