Instructions for use
1. The treatise takes mathematics at its beginning and gives complete proofs. Its reading therefore presupposes, in principle, no particular mathematical knowledge, but only a certain familiarity with mathematical reasoning and a certain power of abstraction. Nevertheless, the treatise is intended more particularly for readers possessing at least a good knowledge of the subjects taught in the first or the first two years of the university.
2. The mode of exposition followed is axiomatic and proceeds most often from the general to the particular. The necessities of the proof require that the chapters follow one another, in principle, in a rigorously fixed logical order. The usefulness of certain considerations will therefore appear to the reader only on reading later chapters, unless he already possesses sufficiently extensive knowledge.
3. The treatise is divided into Books and each Book into chapters. The Books currently published, in whole or in part, are the following:
Theory of Sets designated by E
Algebra — A
General Topology — TG
Functions of a Real Variable — FVR
Topological Vector Spaces — EVT
Integration — INT
Commutative Algebra — AC
Differentiable and Analytic Manifolds — VAR
Lie Groups and Lie Algebras — LIE
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Spectral Theories — TS
Algebraic Topology — TA
In the first six Books (in the order indicated above), each statement calls only upon the definitions and results previously set out in the current chapter or in earlier chapters in the following order: E ; A, chapters I to III ; TG, chapters I to III ; A, chapters IV and following ; TG, chapters IV and following ; FVR ; EVT ; INT. From the seventh Book onwards, the reader will find, possibly at the beginning of each Book or chapter, the precise indication of the other Books or chapters used (the first six Books always being assumed known).
4. However, a few passages are exceptions to the preceding rules. They are placed between two asterisks . In some cases, this is only to facilitate the understanding of the text by examples which refer to facts that the reader may already know elsewhere. Sometimes also, not only the results assumed known in the whole of the current chapter are used, but results proved elsewhere in the treatise. These passages will be used freely in the parts which assume known the chapters where these passages are inserted and the chapters to which these passages refer. The reader will, we hope, be able to verify the absence of any vicious circle.
5. To certain Books (whether published or in preparation) are annexed fascicles of results. These fascicles contain the essential definitions and results of the Book, but no proof.
6. The logical framework of each chapter is constituted by the definitions, axioms and theorems of that chapter; these are what it is principally necessary to retain with a view to what is to follow. The less important results, or those which can easily be recovered from the theorems, appear under the names of “propositions”, “lemmas”, “corollaries”, “remarks”, etc.; those which may be omitted on a first reading are printed in small type. Under the name of “scholium”, one will sometimes find a commentary on a particularly important theo-rem.
To avoid tedious repetitions, it is sometimes agreed to introduce certain notations or certain abbreviations which are valid only within a single chapter or a single paragraph (for
viii example, in a chapter where all rings are commutative, one may agree that the word “ring” always means “commutative ring”). Such conventions are explicitly mentioned at the head of the chapter or paragraph in which they apply.
7. Certain passages are intended to guard the reader against serious errors into which he might fall; these passages are indicated in the margin by the sign ☡ (“dangerous turn”).
8. The exercises are intended, on the one hand, to enable the reader to verify that he has properly assimilated the text; on the other hand, to make him acquainted with results which had no place in the text; the most difficult are marked with the sign .
9. The terminology followed in this treatise has been the object of particular attention. An effort has been made never to depart from the accepted terminology without very serious reasons.
10. An effort has been made to use, without sacrificing the simplicity of the exposition, a rigorously correct language. As far as possible, abuses of language or notation, without which any mathematical text risks becoming pedantic and even unreadable, have been pointed out in passing.
11. The text being devoted to the dogmatic exposition of a theory, bibliographical references will be found only exceptionally; these are sometimes grouped in Historical Notes. The bibliography following each of these Notes most often contains only the original books and memoirs which have had the greatest importance in the evolution of the theory in question; it is in no way intended to be complete.
As for the exercises, it has not generally been judged useful to indicate their provenance, which is very diverse (original memoirs, didactic works, collections of exercises).
12. In the new edition, references to theorems, axioms, definitions, remarks, etc. are given in principle by indicating successively the Book (by the abbreviation corresponding to it in the list given in No.), the chapter and the page where they are found. Within the same Book, the mention of this Book is omitted; for example, in the Book of Algebra,
E, III, p. 32, cor. 3
ix refers to corollary 3 found in the Book of Theory of Sets, chapter III, page 32 of that chapter ;
II, p. 24, prop. 17
refers to proposition 17 of the Book of Algebra, chapter II, page 24 of that chapter.
The results fascicles are designated by the letter R; for example: EVT, R means “results fascicle of the Book on Topological Vector Spaces”.
Since certain Books must be published later in the new edition, references to these Books are made by indicating successively the Book, the chapter, the paragraph and the number where the result in question should be found; for example:
AC, III, § 4, n, cor. of the prop. 6.
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