## Preface to Roger Godement's Analysis I: Convergence, Elementary Functions

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**From the Preface to Analyse Mathématique I.**

It is for the reader that is interested in mathematics for its own sake or as a language of science, and not as a means to achieve the language of questionable technologies, I thought writing it. The only subject here is mathematical analysis as it was and as it has become. I confined myself to functions of a single real or complex variable while limiting the difficulty of the text; this corresponds approximately to the first two volumes, the level imposed by "programmes" in force in France. I also tried to give, in this limited context, demonstrations that could easily extend to the more general situations that the reader will meet later.As for content, I did not hesitate to introduce, sometimes very early, subjects considered relatively advanced - multiple series and unconditional convergence, analytical functions, the definition and immediate properties of Radon measures and distributions, the integrals of semi-continuous functions, Weierstrass elliptic functions, etc. - when they can be exposed without technical complications, reserving the possibility to deepen them in a third volume. I tried to give the reader an idea of the axiomatic construction of set theory hoping he will take Chapter I for what it is - a contribution to the mathematical background - not as a mandatory preliminary to learning analysis. Chapter VII develops, besides the classical theory of Fourier series and integrals, those classical properties of analytic functions or harmonics that can be proved without using the curvilinear Cauchy integral: the simplest results on Fourier series given there are sufficient and I have often taught this less common method; the remainder of the theory will come in Volume III. Conversely, I have not dealt with differential equations. One can learn all about them in myriads of books; classical results of the theory, direct applications of the general principles of the analysis, will not pose any serious problems to a student who has nearly assimilated what is here.

Secondly, I strongly insisted, sometimes using developments in ordinary language, on the basic ideas of analysis and, in some cases, on their historical evolution. I am not, by a long way, an expert on the history of mathematics; some mathematicians, feeling their end is coming, devote their later years; other, younger mathematicians, consider the subject interesting enough to devote a substantial part of their careers; they do very useful work even pedagogically because at twenty years, an age I've been, you just think of rushing straight ahead without looking back and almost always without knowing where we are going: where and when do we learn? I preferred to interest myself for a quarter century in a kind of story for which mathematics does not prepare us but is not without some indirect relationship with it. Nevertheless, I have made some efforts to make the reader understand that the ideas and techniques have evolved and it took one to two centuries before the founders intuitions become perfectly clear ideas based on unassailable arguments, pending broad generalizations in the twentieth century.

Adopting this point of view led me in the largest part of this book to systematically reject perfectly linear presentations, organized like clockwork, and only giving the reader the prevailing or à la mode point of view ... At the risk of establishing a number of classical results several times over, I tried to introduce to the reader several methods of reasoning and, on occasion, to make him understand the need for rigour highlighting questionable reasoning due to mathematicians such as Newton, Bernoulli, Euler and Cauchy. Adopting this view significantly increases the length of the text, but one of the basic principles of N Bourbaki - never save paper - seems to win when teaching students to explore a topic.

The other principle of this same author - to substitute ideas for computations - appears even more commendable to me whenever it can be applied. All the same, one will, inevitably, find calculations in this book; but I have essentially confined myself to those which, inherited from the great mathematicians of the past, form an integral part of the theory and can therefore be considered as ideas.

Numerical calculations have not become pointless: thanks to computers, one can do more and more of them, for better or worse, in all scientific and technical areas that, from medical imaging to prfecting nuclear weapons, use mathematics. One doe the same in certain branches of mathematics too; for example, displaying a large number of curves may open the way to a general theorem or to understanding a topological situation, not to speak of the traditional number theory where numerical experiment always was, and still is, used to formulate or verify conjectures.

This only means that the aim of an exposition of the principles of analysis is not to teach numerical techniques. Moreover, the partisans of applied mathematics, of numerical analysis and of computer science in all universities of the world manifest their imperialist tendencies far too clearly for real mathematicians to take on in their stead a task for which they generally lack both taste and competence.

JOC/EFR November 2014

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