Treatise on Analysis is a translation by Ian G. Macdonald of the nine-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 volumes 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, vol. X, New York-London: Academic Press, MR 0120319 Dieudonné, J. (1963), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII, Paris: Gauthier-Villars, MR 0161945 Dieudonné, J. (1968), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII (2nd ed.), Paris: Gauthier-Villars, MR 0235945 Dieudonné, J. (1969), Foundations of modern analysis., Pure and Applied Mathematics, vol. 10-I (2nd ed.), New York-London: Academic Press, ISBN 978-0122155505, MR 0349288
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, vol. XXXI, Paris: Gauthier-Villars, MR 0235946 Dieudonné, J. (1970), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II, New York-London: Academic Press, MR 0258551 Dieudonné, J. (1976), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II (2nd ed.), New York-London: Academic Press, ISBN 0-12-215502-5, MR 0530406
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, vol. XL11, Paris: Gauthier-Villars, ISBN 2-04-011499-8, MR 0658305
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 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 Marsden, Jerrold E. (1980), "Review: Jean Dieudonné, Treatise on analysis", Bull. Amer. Math. Soc. (N.S.), 3 (1): 719–724, doi:10.1090/s0273-0979-1980-14804-1
