Encyclopaedia of Mathematics: C An updated and annotated translation of the Soviet ‘Mathematical Encyclopaedia’

517 Pages·1988·44.7 MB·English

by  Michiel Hazewinkel (auth.)

ENCYCLOPAEDIA OF MATHEMATICS
Volume 2
ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
Scientific Board
J. F. Adams, S. Albeverio, J. B. Alblas, S. A. Amitsur, I. J. Bakelman, J. W. de Bakker,
C. Bardos, H. Bart, H. Bass, M. Bercovier, M. Berger, E. A. Bergshoeff, E. Bertin, F. Beukers,
A. Beutelspacher, K.-D. Bierstedt, H. P. Boas, J. Bochnak, H. J. M. Bos, B. L. J. Braaksma,
T. P. Branson, D. S. Bridges, A. E. Brouwer, M. G. de Bruin, R. G. Burns, H. Capel, P. Cartier,
C. Cercignani, J. M. C. Clark, Ph. Clement, A. M. Cohen, J. W. Cohen, H. S. M. Coxeter, R. F.
Curtain, M. H. A. Davis, M. V. Dekster, C. Dellacherie, M.Demazure, G. van Dijk,
H. C. Doets, I. Dolgachev, A. Dress, J. J. Duistermaat, D. van Dulst, H. van Duyn, H.Dym,
A. Dynin, M. L. Eaton, W. Eckhaus, J. Eells, P. van Emde Boas, H. Engl, G. Ewald,
V. I. Fabrikant, A. Fasano, M. Fliess, R. M. Fossum, M. Freidlin, B. Fuchssteiner, G. B. M.
van der Geer, R. D. Gill, V. V. Goldberg, J. de Graaf, J. Grasman, P. A. Griffith,
A. W. Grootendorst, L. Gross, P. Gruber, E. J. Hannan, K. P. Hart, A. van Harten, G. Heckman,
A. J. Hermans, W. H. Hesselink, C. C. Heyde, M. W. Hirsch, K. H. Hofmann, A. T. de Hoop,
P. J. van der Houwen, N. M. Hugenholtz, J. R. Isbell, A.Isidori, E. M. de Jager, D. Johnson,
P. T. Johnstone, D. Jungnickel, M. A. Kaashoek, V. Kac, W. L. J. van der Kallen, D. Kanevsky,
Y. Kannai, H. Kaul, E. A. de Kerf, W. Klingenberg, T. Kloek, J. A. C. Kolk, G. Komen,
T. H. Koornwinder, E. Kranakis, L. Krop, B. Kuperschmidt, H. A. Lauwerier, J. van Leeuwen,
J. Lennox, H. W. Lenstra Jr., J. K. Lenstra, H. Lenz, M. Levi, J. Lindenstrauss, J. H. van Lint,
F. Linton, A. Liulevicius, M. Livshits, W. A. J. Luxemburg, R. M. M. Mattheij, L. G. T. Meer
tens, P. Mekenkamp, A. R. Meyer, J. van Mill, I. Moerdijk, H. Neunzert, G. Y. Nieuwland,
G. J. Olsder, B. 0rsted, F. van Ostayen, B. Pareigis, K. R. Parthasarathy, I. I. Piatetskii-Shapiro,
H. G. J. Pijls, N. U. Prabhu, G. B. Preston, E. Primrose, A. Ramm, C. M. Ringel, J. B. T. M.
Roerdink, K. W. Roggenkamp, G. Rozenberg, W. Rudin, F. H. Ruymgaart, S. N. M.
Ruysenaars, A. Salam, A. Salomaa, P. Saunders, J. P. M. Schalkwijk, C. L. Scheffer, R.
Schneider, J. A. Schouten, F. Schurer, I. A. Segal, J. J. Seidel, A. Shenitzer, V. Snaith,
T. A. Springer, J. H. M. Steenbrink, J. D. Stegeman, F. W. Steutel, P. Stevenhagen, I. Stewart,
L. Streit, K. Stromberg, L. G. Suttorp, D. Tabak, F. Takens, R. J. Takens, N. M. Temme,
S. H. Tijs, B. Trakhtenbrot, N. S. Trudinger, L. N. Vaserstein, M. L. J. van der Vel, F. D.
Veldkamp, W. Vervaat, P. M. B. Vitanyi, H. A. van der Vorst, J. de Vries, F. Waldhausen,
B. Wegner, J. J. O. O. Wiegerinck, J. Wiegold, 1. C. Willems, J. M. Wills, B. de Wit,
S. A. Wouthuysen, S. Yuzvinskii, L. Zalcman
ENCYCLOPAEDIA
OF
MATHEMATICS
Volume 2
c
An updated and annotated translation of the Soviet
'Mathematical Encyclopaedia'
KLUWER ACADEMIC PUBLISHERS
Dordrecht / Boston / London
Library of Congress Cataloging-in-Publication Data
Matematicheskaia entsiklopediia. English.
Encyclopaedia of mathematics.
1. Mathematics--Dictionaries. I. Hazewinkel, Michie!. II. Title.
QA5.M3713 1987 510'.3'21 87-26437
ISBN-13: 978-94-009-6002-2 e-lSBN-13: 978-94-009-6000-8
001: 10.1007/978-94-009-6000-8
Published by Kluwer Academic Publishers,
P.O Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
All Rights Reserved
© 1988 by Kluwer Academic Publishers
Softcover reprint of the hardcover 15t edition 1988
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
induding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
SOVIET MATHEMATICAL ENCYCLOPAEDIA
Editor-in-Chief
I. M. Vinogradov
Editorial Board
S. I. Adyan, P. S. Aleksandrov, N. S. Bakhvalov, A. V. Bitsadze, V. I. Bityutskov (Deputy
Editor-in-Chief), L. N. Bol'shev, A. A. Gonchar, N. V. Efimov, V. A. Il'in, A. A. Karatsuba,
L. D. Kudryavtsev, B. M. Levitan, K. K. Mardzhanishvili, E. F. Mishchenko, S. P. Novikov,
E. G. Poznyak, Yu. V. Prokhorov (Deputy Editor-in-Chief), A. I. Shirshov, A. G. Sveshnikov,
A. N. Tikhonov, P. L. UI'yanov, S. V. YablonskiI
Translation Arrangements Committee
V. I. Bityutskov, R. V. Gamkrelidze, Yu. V. Prokhorov
'Soviet Encyclopaedia' Publishing House
PREFACE
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathematics.
It is a translation with updates and editorial comments of the Soviet Mathematical En
cyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in
1977 - 1985. The annotated translation consists of ten volumes including a special index
volume.
There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type
articles dealing with the various main directions in mathematics (where a rather fine subdivision
has been used). The main requirement for these articles has been that they should give a
reasonably complete up-to-date account of the current state of affairs in these areas and that they
should be maximally accessible. On the whole, these articles should be understandable to mathe
matics students in their first specialization years, to graduates from other mathematical areas
and, depending on the specific subject, to specialists in other domains of science, engineers and
teachers of mathematics. These articles treat their material at a fairly general level and aim to
give an idea of the kind of problems, techniques and concepts involved in the area in question.
They also contain background and motivation rather than precise statements of precise theorems
with detailed definitions and technical details on how to carry out proofs and constructions.
The second kind of article, of medium length, contains more detailed concrete problems,
results and techniques. These are aimed at a smaller group of readers and require more back
ground expertise. Often these articles contain more precise and refined accounts of topics and
results touched upon in a general way in the first kind of article.
Finally, there is a third kind of article: short (reference) definitions.
Practically all articles (all except a few of the third kind) contain a list of references by means
of which more details and more material on the topic can be found. Most articles were specially
written for the encyclopaedia and in such cases the names of the original Soviet authors are
mentioned. Some articles have another origin such as the Great Soviet Encyclopaedia
(Bol'shaya Sovetskaya Entsiklopediya or BSE).
Communication between mathematicians in various parts of the world has certainly greatly
improved in the last decennia. However, this does not mean that there are so-to-speak 'one-to
one onto' translations of the terminology, concepts and tools used by one mathematical school
to those of another. There also are varying traditions of which questions are important and
which not, and what is considered a central problem in one tradition may well be besides the
point from the point of view of another. Even for well-established areas of mathematical in-
4uiry, terminology varies across languages and even within a given language domain. Further, a
concept, theorem, algorithm, ... , which is associated with one proper name within one tradition
may well have another one in another, especially if the result or idea in question was indeed
discovered independently and more-or-Iess simultaneously. Finally, mathematics is a very
dynamic science and much has happened since the original articles were finalized (mostly
around 1977). This made updates desirable (when needed). All this, as well as providing addi
tional references to Western literature when needed, meant an enormous amount of work for the
PREFACE
board of experts as a whole; some indeed have done a truly impressive amount of work. I must
stress though that I am totally responsible for what is finally included and what is not of all the
material provided by the members of the board of experts.
Many articles are thus provided with an editorial comment section in a different and some
what smaller typeface. In particular, these annotations contain additional material, amplifica
tions, alternative names, additional references, .... Modifications, updates and other extra
material provided by the original Soviet authors (not a rare occurrence) have been incorporated
in the articles themselves.
The final (lO-th) volume of the ENCYCLOPAEDIA OF MATHEMATICS will be an index volume.
This index will contain all the titles of the articles (some 6600) and in addition the names of all
the definitions, named theorems, algorithms, lemmas, scholia, constructions, ... , which occur
in the various articles. This includes, but is by no means limited to, all items which are printed in
bold or italic. Bold words or phrases, by the way, always refer to another article with (precisely)
that title.
All articles have been provided with one or more AMS classification numbers according to
the 1980 classification scheme (not, for various reasons, the 1985 revision), as have all items
occurring in the index. A phrase or word from an article which is included in the index always
inherits all the classification numbers of the article in question. In addition, it may have been
provided with its own classification numbers. In the index volume these numbers will be listed
with the phrase in question. Thus e.g. the Quillen - Suslin theorem of algebraic K-theory will
have its own main classification numbers (these are printed in bold; in this case that number is
18F25) as well as a number of others, often from totally different fields, pointing e.g. to parts of
mathematics where the theorem is applied, or where there occurs a problem related to it (in this
case e.g. 93D15). The index volume will also contain the inversion of this list which will, for
each number, provide a list of words and phrases which may serve as an initial description of the
'content' of that classification number (as far as this ENCYCLOPAEDIA is concerned). For more
details on the index volume, its structure and organisation, and what kind of things can be done
with it, cf. the (future) special preface to that volume.
Classifying articles is a subjective matter. Opinions vary greatly as to what belongs where and
thus this attempt will certainly reflect the tastes and opinions of those who did the classification
work. One feature of the present classification attempt is that the general basic concepts and
definitions of an area like e.g. 55N (Homology and Cohomology theories) or 601 (Markov
processes) have been assigned classification numbers like 55NXX and 60JXX if there was no
finer classification number different from ... 99 to which it clearly completely belongs.
Different parts of mathematics tend to have differences in notation. As a rule, in this
ENCYCLOPAEDIA in a given article a notation is used which is traditional in the corresponding
field. Thus for example the (repeated index) summation convention is used in articles about
topics in fields where that is traditional (such as in certain parts of differential geometry (tensor
geometry» and it is not used in other articles (e.g. on summation of series). This pertains espe
cially to the more technical articles.
For proper names in Cyrillic the British Standards Institute transcription system has been used
(cf. Mathematical Reviews). This makes well known names like S.N. Bernstein come out as
Bernshtein.
In such cases, especially in names of theorems and article titles, the traditional spelling has
been retained and the standard transcription version is given between brackets.
Ideally an encyclopaedia should be complete up to a certain more-or-Iess well defined level of
detail. In the present case I would like to aim at the completeness level whereby every theorem,
viii
PREFACE
concept, definition, lemma, construction which has a more-or-Iess constant and accepted name
by which it is referred to by a recognizable group of mathematicians occurs somewhere, and can
be found via the index. It is unlikely that this completeness ideal will be reached with this
present ENCYCLOPAEDIA OF MATHEMATICS, but it certainly takes substantial steps in this direc
tion. Everyone who uses this ENCYCLOPAEDIA and finds items which are not covered, which, he
feels, should have been included, is invited to inform me about it. When enough material has
come in this way supplementary volumes will be put together.
The ENCYCLOPAEDIA is alphabetical. Many titles consist of several words. Thus the problem
arises how to order them. There are several systematic ways of doing this of course, for instance
using the first noun. All are unsatisfactory in one way or another. Here an attempt has been
made to order things according to words or natural groups of words as they are daily used in
practice. Some sample titles may serve to illustrate this: Statistical mechanics, mathematical
problems in; Lie algebra; Free algebra; Associative algebra; Absolute continuity; Abstract
algebraic geometry; Boolean functions, normal forms of. Here again taste plays a role (and
usages vary). The index will contain all permutations. Meanwhile it will be advisable for the
reader to try out an occasional transposition himself. Titles like K-theory are to be found under
K, more precisely its lexicographic place is identical with 'K theory', i.e. '-' = 'space' and
comes before all other symbols. Greek letters come before the corresponding Latin ones, using
the standard transcriptions. Thus X2-distribution (chi-squared distribution) is at the beginning of
the letter C. A * as in C* -algebra and *-regular ring is ignored lexicographically. Some titles
involve Greek letters spelled out in Latin. These are of course ordered just like any other
'ordinary'title.
This volume has been computer typeset using the (Unix-based) system of the CWI, Amster
dam. The technical (mark-up-Ianguage) keyboarding was done by Rosemary Daniels, Chahrzade
van 't Hoff and Joke Pesch. To meet the data-base and typesetting requirements of this
ENCYCLOPAEDIA substantial amounts of additional programming had to be done. This was done
by Johan Wolleswinkel. Checking the translations against the original texts, and a lot of desk
editing and daily coordination was in the hands of Rob Hoksbergen. All these persons, the
members of the board of experts, and numerous others who provided information, remarks and
material for the editorial comments, I thank most cordially for their past and continuing efforts.
The original Soviet·version had a printrun of 150,000 and is completely sold out. I hope that
this annotated and updated translation will tum out to be comparably useful.
Bussum, August 1987 MICHIEL HAZEWINKEL
_ _ c _ _
x
2 -DISTRIBUTION - See 'Chi-squared' distribution. C· -algebra of the form L(H). Any closed two-sided
ideal [ in a C'-algebra is self-adjoint (thus [ is a C'
AMS 1980 Subject Classification: 62EXX
subalgebra of A), and the quotient algebra A / [,
endowed with the natural linear operations, multiplica
c* -ALGEBRA - A Banach algebra A over the field tion, involution, and quotient space norm, is a C'
of complex numbers, with an involution x~x', xEA, algebra. The set K (H) of completely-continuous linear
such that the norm and the involution are connected by operators on a Hilbert space H is a closed two-sided
the relation II x· x II = II X 112 for any element x EA. ideal in L (H). If A is a C· -algebra and A is the algebra
C' -algebras were introduced in 1943 [1] under the with involution obtained from A by addition of a unit
A
name of totally regular rings; they are also known under element, there exists a unique norm on which con
A
the name of B' -algebras. The most important examples verts into a C· -algebra and which extends the norm
of C' -algebras are: on A. Moreover, the operations of bounded direct sum
1) The algebra Co(X) of continuous complex-valued and tensor product [3], [4] have been defined for C'
functions on a locally compact Hausdorff space X algebras.
which tend towards zero at infinity (i.e. continuous As in all symmetric Banach algebras with involution,
functions f on X such that, for any £>0, the set of in a C· -algebra A it is possible to define the following
points x EX which satisfy the condition If (x) I ~£ is subsets: the real linear space Ah of Hermitian elements;
compact in X); Co(X) has the uniform norm the set of normal elements; the multiplicative group U
of unitary elements (if A contains a unit element); and
II/II = ~~~ I/(x) I· the set A + of positive elements. The set A + is a
The involution in Co(X) is defined as transition to the closed cone in Ah, A + n(-A)+ ={O}, A + -A + =Ah'
complex-conjugate function: r(x)= f(x). Any com and the cone A + converts Ah into a real ordered vector
mutative C' -algebra A is isometrically and symmetri space. If A contains a unit element 1, then 1 is an inte
cally isomorphic (i.e. is isomorphic as a Banach algebra rior point of the cone A + CAh. A linear functional f
A with involution) to the C' -algebra Co(X), where X is on A is called positive if f(x)~O for all x EA +; such a
the space of maximal ideals of A endowed with the functional is continuous. If x EB, where B is a C'
Gel'fand topology [1], [2], [3]. subalgebra of A, the spectrum of x in B coincides with
2) The algebra L (H) of all bounded linear operators the spectrum of x in A. The spectrum of a Hermitian
on a Hilbert space H, considered with respect to the element is real, the spectrum of a unitary element lies
ordinary linear operations and operator multiplication. on the unit circle, and the spectrum of a positive ele
The involution in L (H) is defined as transition to the ment is non-negative. A functional calculus for the
adjoint operator, and the norm is defined as the ordi normal elements of a C· -algebra has been constructed.
nary operator norm. Any C· -algebra A has an approximate unit, located in
A subset MeA is said to be self-adjoint if M = M' , the unit ball of A and formed by positive elements of
where M'={x': xEM}. Any closed self-adjoint A. If [, J are closed two-sided ideals in A, then (I +J)
subalgebra B of a C'-algebra A is a C'-algebra with is a closed two-sided ideal in A and
respect to the linear operations, multiplication, involu (/ +J)+ =[+ +J +. If / is a closed two-sided ideal in
tion, and norm taken from A; B is said to be a C' J and J is a closed two-sided ideal in A, then [ is a
subalgebra of A. Any C'-algebra is isometrically and closed two-sided ideal in A. Any closed two-sided ideal
symmetrically isomorphic to a C· -subalgebra of some is the intersection of the primitive two-sided ideals in