Treatise on analysis

Treatise on analysis is a translation of the 9-volume work Éléments d'analyse on mathematical analysis by Jean Dieudonné, and is an expansion of his textbook Foundations of modern analysis. It is a successor to the various Cours d'Analyse by Augustin-Louis Cauchy, Camille Jordan, and Édouard Goursat.

Contents and publication history

Volume I

The first volume was originally a stand-alone graduate textbook with a different title. It was first written in English and later translated into French, unlike the other volume which were first written in French. It has been republished several times and is much more common than the later volumes of the series.

The contents include

• Chapter I: sets
• Chapter II Real numbers
• Chapter III Metric spaces
• Chapter IV The real line
• Chapter V Normed spaces
• Chapter VI Hilbert spaces
• Chapter VII Spaces of continuous functions
• Chapter VIII Differential calculus (This uses the Cauchy integral rather than the more common Riemann integral of functions.)
• Chapter IX Analytic functions (of a complex variable)
• Chapter X Existence theorems (for ordinary differential equations)
• Chapter XI Elementary spectral theory

• Dieudonné, J. (1960), Foundations of modern analysis, Pure and Applied Mathematics, X, New York-London: Academic Press, https://archive.org/details/FoundationsOfModernAnalysis_578 
• Dieudonné, J. (1963), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, XXVIII, Paris: Gauthier-Villars 
• Dieudonné, J. (1968), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, XXVIII (2nd ed.), Paris: Gauthier-Villars 
• Dieudonné, J. (1969), Foundations of modern analysis., Pure and Applied Mathematics, 10-I (2nd ed.), New York-London: Academic Press, ISBN 978-0122155505 

Volume II

The second volume includes

• Chapter XII Topology and topological algebra
• Chapter XIII Integration
• Chapter XIV Integration in locally compact groups
• Chapter XV Normed algebras and spectral theory

• Dieudonné, J. (1968), Éléments d'analyse. Tome II: Chapitres XII à XV, Cahiers Scientifiques, XXXI, Paris: Gauthier-Villars 
• Dieudonné, J. (1970), Treatise on analysis. Vol. II, Pure and Applied Mathematics, 10-II, New York-London: Academic Press 
• Dieudonné, J. (1976), Treatise on analysis. Vol. II, Pure and Applied Mathematics, 10-II (2nd ed.), New York-London: Academic Press, ISBN 0-12-215502-5 

Volume III

The third volume includes chapter XVI on differential manifolds and chapter XVII on distributions and differential operators.

Volume IV

The fourth volume includes

• Chapter XVIII Differential systems
• Chapter XIX Lie groups
• Chapter XX Riemannian geometry

Volume V

Volume V consists of chapter XXI on compact Lie groups.

Volume VI

Volume VI consists of chapter XXII on harmonic analysis (mostly on locally compact groups)

Volume VII

Volume VII consists of the first part of chapter XXIII on linear functional equations. This chapter is considerably more advanced than most of the other chapters.

Volume VIII

Volume VIII consists of the second part of chapter XXIII on linear functional equations.

Volume IX

Volume IX contains chapter XXIV on elementary differential topology. Unlike the earlier volumes there is no English translation of it.

• Dieudonné, J. (1982), Éléments d'analyse. Tome IX. Chapitre XXIV, Cahiers Scientifiques, XL11, Paris: Gauthier-Villars, ISBN 2-04-011499-8 

Volume X

Dieudonne planned a final volume containing chapter XXV on nonlinear problems, but this was never published.

References

• Nachbin, Leopoldo (1961), "Review: J. Dieudonné, Foundations of Modern Analysis", Bull. Amer. Math. Soc. 67 (3): 246–250, doi:10.1090/s0002-9904-1961-10566-1, http://projecteuclid.org/euclid.bams/1183524143 
• Frank, Peter (1960), "Book reviews: Foundations of Modern Analysis. J. Dieudonné. Academic Press, New York, 1960", Science 132 (3441): 1759, doi:10.1126/science.132.3441.1759-a 
• {{citation|journal=Bull. Amer. Math. Soc. (N.S.)

|volume =3|issue= 1|year= 1980|pages= 719–724 |title=Review: Jean Dieudonné, Treatise on analysis