Analyse I: Théorie des ensembles et topologie. Avec la collaboration de K. Zizi. (French) Zbl 0872.54001

Schwartz, Laurent

Collection Enseignement des Sciences. 42. Paris: Hermann, Éditeurs des Sciences et des Arts. 404 p. (1991).

This is the first of four volumes of the now completely revised and considerably enhanced edition of the famous “Cours d’analyse”, first published in 1967 (see Zbl 0171.01301). This first volume consists of two chpaters: I. Set theory, and II. Topology.
Contents: Chapter I. Set theory: \(\S 1\) Some elements of classical logic; \(\S 2\) Set theory – the five primary axioms; \(\S 3\) Mappings, family, product of a family of sets, axiom of choice; \(\S 4\) The natural numbers – the axiom of infinity; \(\S 5\) Quotient sets; \(\S 6\) Ordered sets; \(\S 7\) Infinite sets – operations on infinite sets; \(\S 8\) Ordinal and cardinal numbers.
Chapter II. Topology: \(\S 1\) Metric spaces; \(\S 2\) Topological spaces; \(\S 3\) Continuous and semi-continuous functions – homeomorphisms; \(\S 4\) Metric spaces and topological spaces; \(\S 5\) Compact spaces – elementary properties; \(\S 6\) Convergence, limits, sequences and filters; \(\S 7\) Properties of continuous functions on a compact space; \(\S 8\) Locally compact spaces; \(\S 9\) Connected spaces, arc-connected spaces, locally connected spaces; \(\S 10\) Complete metric spaces; \(\S 11\) Elementary theory of normed linear spaces and Banach spaces; \(\S 12\) Series in normed linear spaces; \(\S 13\) Function spaces – pointwise and uniform convergence; \(\S 14\) Elementary spectral theory (including the Gel’fand-Najmark theorem and the Bochner-Raikov theorem); \(\S 15\) Infinite products of numbers or of real or complex functions.
Numerous examples are given, but exercises are missing. The bibliography contains 34 titles, and the book is completed by a terminological index.

MSC:

54-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
03-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations

Keywords:

set theory; topology; function spaces; normed linear spaces; Banach spaces

Citations:

Zbl 0171.01301