×

Andreĭ Nikolaevich Tikhonov. Collection of scientific works. In ten volumes. Vol. 1: Mathematics. Part 1: Topology, functional analysis 1925–1966. Equations with small parameter and asymptotic methods 1948–1985. Mathematical physics 1946–1975. Edited by T. A. Sushkevich and V. F. Butuzov. (Андрей Николаевич Тихонов. Собрание научных трудов в десяти томах. Том I. Математика. Часть 1: Топология, функциональный 1925–1966. Уравнения с малым параметром и асимптотические методы 1948–1985. Математическая физика 1946–1975.) (Russian, English) Zbl 1262.01025

Klassiki Nauki. Moskva: Nauka (ISBN 978-5-02-036160-7/hbk; 978-5-02-036158-4/set). 636 p. (2012).
This is Part 1 of Volume 1 of the Collection of Scientific Works of Andreĭ Nikolaevich Tikhonov (1936–1993) (Part 2 of Volume 2 and Part 1 of Volume 3 have been already published, see [Zbl 1200.01055; Zbl 1184.01038]). This Part 1, which starts the whole collection, begins with an article on Tikhonov by V. A. Iliin followed by an article on his scientific work and achievements (here Tikhonov’s part in the development of the Soviet hydrogen bomb is recalled) together with a plan of the collection and a preface, the two latter by the Editorial Committee. The proper part of this Part 1 consists of three sections: seventeen papers in topology and functional analysis from the years 1925–1966 (preceded by two articles, one by P. S. Aleksandrov and S. V. Fomin on Tikhonov’s achievements from 1966 and another one by P. S. Aleksandrov on Tikhonov’s discoveries in topology from 1966), thirteen papers in differential equations with small parameters and asymptotic methods from the years 1948–1985, and five papers in mathematical physics from the years 1946–1975.
In topology, Tikhonov’s earliest area off interest, he has made some major inventions still as a teenager. He defined a topology on Cartesian products of arbitrarily many topological spaces and proved the important theorem that such a product of compact spaces is compact itself. To him belongs also the definition of a completely regular topological space and the theorem that any completely regular space with a countable basis is metrizable. Extending his interests to functional analysis he proved the theorem that if \(X\) is a compact subspace of a locally convex linear space, then any continuous mapping of \(X\) into itself has a fixed point. The theorem showed the value of the class of locally convex linear spaces and marked a shift of Tikhonov’s interests towards the theory of that class and applications of his theorem to differential equations, particularly those originated in mathematical physics.
The edition seems to be oriented to a Russian reader with a command of English: each paper originally published in French or German is reprinted in its original version and then translated into Russian, while papers originally published in Russian or English are reprinted as they were (papers in English are not translated). Most papers are in Russian (in this volume there are 23 papers in Russian only, out of 35).

MSC:

01A75 Collected or selected works; reprintings or translations of classics
01A60 History of mathematics in the 20th century
54-03 History of general topology
46-03 History of functional analysis
35-03 History of partial differential equations
PDFBibTeX XMLCite