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© 2004

Analysis I

Convergence, Elementary functions

  • Prefers ideas to calculations

  • Explains the ideas without parsimony of words

  • Based on 35 years of teaching at Paris University

  • Blends mathematics skilfully with didactical and historical considerations

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XXI
  2. Roger Godement
    Pages 1-43
  3. Roger Godement
    Pages 45-185
  4. Roger Godement
    Pages 187-324
  5. Back Matter
    Pages 425-434

About this book

Introduction

Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.

Keywords

Banach Space Fourier Fourier series analytic functions calculus convergence derivations integrals intermediate value theorem logarithm

Authors and affiliations

  1. 1.Département de MathématiquesUniversité Paris VIIParis Cedex 05France

About the authors

Roger Godement (October 1, 1921 - July 21, 2016) is known for his work in functional analysis, and also his expository books. He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. He also wrote texts on Lie groups, abstract algebra and mathematical analysis.

Bibliographic information

  • Book Title Analysis I
  • Book Subtitle Convergence, Elementary functions
  • Authors Roger Godement
  • Series Title Universitext
  • DOI https://doi.org/10.1007/978-3-642-18491-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-05923-3
  • eBook ISBN 978-3-642-18491-8
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XXI, 430
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Additional Information Original French edition published by Springer-Verlag, 1998
  • Topics Analysis
    Real Functions
  • Buy this book on publisher's site
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Reviews

From the reviews of the original French edition:

"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.
J. Mawhin in Zentralblatt Mathematik (1999)

From the reviews:

"Analysis I is the translation of the first volume of Godement’s four-volume work Analyse Mathématique, which offers a development of analysis more or less from the beginning up to some rather advanced topics. … the organization of the material is radically different … . It would … make excellent supplementary reading for honors calculus courses." (Gerald B. Folland, SIAM Review, Vol. 47 (3), 2005)

"A book on analysis that is quite different from all other books on this subject. … for those who essentially know the material (the level of an average graduate student, say), and who are interested in mathematics will certainly love reading it. Those who lecture this material may find a lot of inspiration to make their lessons entertaining." (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, Vol. 12 (2), 2005)

"Analysis I is an English translation of the first volume of a four-volume work. Analysis I consists of a spirally organized, organic, non-linear treatment of the introductory areas of ‘mathematical analysis as it was and as it has become’. It is infused with some excellent, sensitive appreciations of the work of pioneers … and reads as a heady blend of both classical concerns and modern refinements, often illuminated by a variety of approaches." (Nick Lord, The Mathematical Gazette, March, 2005)