''Treatise on Analysis'' is a translation by

Ian G. Macdonald Ian Grant Macdonald (11 October 1928 – 8 August 2023) was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinator ...

of the nine-volume work ''Éléments d'analyse'' on

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous ...

, 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 Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...

, and

Édouard Goursat Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his ''Cours d'analyse mathématique'', which appeared in the first decade of the twentieth century. It s ...

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 space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s *Chapter IV The real line *Chapter V

Normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...

s *Chapter VI

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

s *Chapter VII Spaces of continuous functions *Chapter VIII

Differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

(This uses the Cauchy integral rather than the more common

Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...

of functions.) *Chapter IX

Analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

s (of a complex variable) *Chapter X Existence theorems (for

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s) *Chapter XI Elementary

spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...

* * * *

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 * * *

Volume III

The third volume includes chapter XVI on

differential manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

s and chapter XVII on

distributions Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

and

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

Volume IV

The fourth volume includes *Chapter XVIII Differential systems *Chapter XIX

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s *Chapter XX

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

Volume V

Volume V consists of chapter XXI on compact Lie groups.

Volume VI

Volume VI consists of chapter XXII on

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

(mostly on

locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...

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 In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

. Unlike the earlier volumes there is no English translation of it. *

Volume X

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

References

* * *{{citation, journal=Bull. Amer. Math. Soc. (N.S.) , volume =3, issue= 1, year= 1980, pages= 719–724 , title=Review: Jean Dieudonné, ''Treatise on analysis'' , first=Jerrold E., last= Marsden, authorlink=Jerrold E. Marsden, url=http://projecteuclid.org/euclid.bams/1183546476, doi=10.1090/s0273-0979-1980-14804-1, doi-access=free Mathematics books Mathematical analysis Treatises