Tensor Space s and Exterio r Algebra Recent Titles in This Series

108 Take o Yokonuma, Tenso r space s and , 199 2 107 B . M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selecte d problem s in real analysis, 199 2 106 G.-C . Wen, Conforma l mapping s and boundary value problems, 199 2 105 D . R. Yafaev, Mathematica l scatterin g theory: Genera l theory, 199 2 104 R . L. Dobrushin, R. Kotecky, an d S. Shlosman, Wulf f construction : A global shape fro m local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and their applications, 199 2 102 A . M. Il'in, Matchin g o f asymptotic expansions of solution s of boundary valu e problems, 199 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory of differential equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbits i n invariant theory, 199 2 99 Nori o Shimakura, Partia l differential operator s of elliptic type, 199 2 98 V . A. Vassiliev, Complement s o f discriminants o f smooth maps: Topolog y and applications, 199 2 97 Itir o Tamura, Topolog y of foliations: A n introduction, 1 992 96 A . I. Markushevich, Introductio n to the classical theory of Abelian functions, 199 2 95 Guangchan g Dong, Nonlinea r partial s of second order, 199 1 94 Yu . S. Il'yashenko, Finitenes s theorems for limit cycles , 199 1 93 A . T. Fomenko and A. A. Tuzhilin, Element s of the geometry and of minima l surfaces i n three-dimensional space , 199 1 92 E . M. Nikishin and V. N. Sorokin, Rationa l approximations and orthogonality, 199 1 91 Mamon i Mimura and Hirosi Toda, Topolog y of Li e groups, I and II, 199 1 90 S . L. Sobolev, Som e applications o f functiona l analysi s in mathematical , third edition, 199 1 89 Valeri i Y. Kozlov and Dmitrii V. Treshchev, Billiards : A genetic introduction t o the dynamics of systems with impacts, 199 1 88 A . G. Khovanskii, Fewnomials , 199 1 87 Aleksand r Robertovic h Kemer, Ideal s of identities of associative algebras, 199 1 86 V . M. Kadets and M. I. Kadets, Rearrangement s o f series in Banach spaces , 199 1 85 Miki o Ise and Masaru Takeuchi, Li e groups I, II, 199 1 84 Da o Trong Thi an d A. T. Fomenko, Minima l surfaces, stratifie d multivarifolds , an d th e Plateau problem, 199 1 83 N . I. Portenko, Generalize d diffusio n processes , 199 0 82 Yasutak a Sibuya, Linea r differential equation s in the complex domain: Problem s of analytic continuation, 199 0 81 I . M. Gelfand an d S. G . Gindikin, Editors, Mathematica l problems o f tomography, 199 0 80 Junjir o Noguchi an d Takushiro Ochiai, Geometri c function theor y i n several comple x variables, 199 0 79 N . I. Akhiezer, Element s of the theory o f elliptic functions, 199 0 78 A . V. Skorokhod, Asymptoti c methods of the theory of stochastic differential equations , 1989 77 V . M. Filippov, Variationa l principles for nonpotentia l operators , 198 9 76 Philli p A. Griffiths, Introductio n to algebraic curves, 198 9 75 B . S. Kashin and A. A. Saakyan, Orthogona l series, 198 9 74 V . I. Yudovich, Th e linearization metho d i n hydrodynamical stabilit y theory, 198 9 (Continued in the back of this publication) 10.1090/mmono/108

Translations o f MATHEMATICAL MONOGRAPHS

Volume 10 8

Tensor Space s and Exterio r Algebr a

Takeo Yokonuma

Translated b y Takeo Yokonum a

o America n Mathematical Societ y i? Providence , Rhod e Islan d T-vw&mttmft

TENSORU KUUKA N T O GAISEKIDAISU U (Tensor Spaces and Exterior Algebra) by Takeo Yokonuma Copyright © 197 7 by Takeo Yokonuma Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo in 1977 Translated fro m th e Japanese by Takeo Yokonuma

2000 Subject Classification. Primar y 15A69 , 47-XX ; Secondary 15A75 , 81-XX .

ABSTRACT. Thi s book provides an introduction to and related topics. Th e book begins with definitions o f th e basic concepts of th e theory; s of vecto r spaces, tensors, tensor algebras, and exterior algebra. Thei r properties are then studied and applications given. Algebraic systems with bilinear multiplication are introduced i n the final chapter. I n partic- ular, the theory of replica s of Chevalley and several properties of Li e algebras that follow fro m this theory are presented.

Library of Congress Cataloging-in-Publication Data Yokonuma, Takeo, 1939- [Tensoru kuukan to gaisekidaisuu. English ] Tensor spaces and exterior algebra/Takeo Yokonuma; translated by Takeo Yokonuma. p. cm.—(Translations of mathematical monographs, ISSN 0065-9282; 108) of: Tensor u kuukan to gaisekidaisuu. Includes bibliographical reference s and index. ISBN 0-8218-4564-0 1. . 2 . Tensor products. I . Title. II . Series. QA199.5.Y6513 199 2 92-1672 1 512'.57-dc20 CI P

AMS softcover ISBN 978-0-8218-2796-3

Copyright ©1992 b y the American Mathematical Society . Al l rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is arid-free and falls within the guidelines established to ensure permanence and durability. © This publication was typeset using Aj^S-T^i, the American Mathematical Society's T^ X macr o system. Visit the AM S home page at URL : http://www.ams.org/ 10 9 8 7 6 5 4 3 2 0 6 05 04 03 02 01 Contents

Preface t o the Englis h Edition vi i Preface i x Chapter I. Definition o f Tensor Products 1 §1. Preliminaries 1 §2. Definition o f bilinear mappings 2 §3. Linearization o f bilinear mapping s 4 §4. Definition o f tensor products 7 §5. Properties o f tensor products 8 §6. Multilinear mapping s and tensor products o f more than tw o vector spaces 1 1 §7. Tensor products o f linear mapping s 1 4 §8. Examples of tensor product space s 1 8 §9. Construction o f tensor products with generators and relations 2 4 §10. Tensor products o f i?- s 2 6 Exercises 3 0 Chapter II. Tensors an d Tensor Algebras 3 3 §1. Definition an d examples o f tensors 3 3 §2. Properties o f tensor spaces 3 6 §3. Symmetric tensors and alternating tensors 4 3 §4. Tensor algebras and their properties 5 4 §5. Symmetric algebras and their properties 6 2 §6. Definition o f relativ e tensors ( , tenso r density) 6 8 Exercises 7 3 Chapter III. Exterior Algebr a and it s Applications 7 7 §1. Definition o f exterior algebra and it s properties 7 7 §2. Applications to 8 3 §3. Inner (interior ) products o f exterior algebras 8 8 §4. Applications to geometry 9 1 Exercises 9 7 VI CONTENTS

Chapter IV . Algebraic Systems with Bilinear Multiplication. Li e Algebras 10 1 §1. Algebraic system s with bilinear multiplication 10 1 §2. Replicas of matrices 10 5 §3. Properties of Lie algebras 11 4 Exercises 12 5 References fo r th e Englis h Edition 12 7 Subject Inde x 129 Preface t o the English Editio n

This is a translation o f m y work originally published b y Iwanami Shoten , as a volume in their Lecture Serie s on linear algebra . We assume, therefore, tha t reader s are familiar wit h severa l fundamenta l concepts of linear algebra such as vector spaces, matrices, determinants, etc., though w e review som e o f these concepts i n the text. W e have made som e changes in the references . On thi s occasion , I woul d lik e to expres s m y gratitude t o Professo r Na - gayoshi Iwahori for giving me encouragement and many valuable suggestions during the preparatio n o f th e origina l wor k an d als o to Mr. Hide o Ara i o f Iwanami Shote n fo r continue d support . I am grateful t o the American Mathematical Societ y and the staff for thei r effort i n publishin g thi s Englis h edition . I als o than k Kazunar i Nod a an d Nami Yokonuma fo r their excellent typing.

Takeo Yokonum a December 199 1 This page intentionally left blank Preface

The subject matter of the present book is generally called multilinear alge- bra; w e shall discus s mainl y tensors an d relate d concepts . Th e readers ma y recall severa l importan t tensor s tha t ar e use d i n differentia l geometry , me - chanics, electromagnetics, and s o on. O n the other hand, i t is generally sai d that tensor s ar e difficul t t o understand . On e o f th e reason s fo r thi s i s tha t formerly tensor s and tensor fields (mappings whose values are tensors) wer e not distinguished , an d tenso r fields were discusse d withou t definin g tensor s in advance. ( ) I n fact , reader s should be aware that sometime s tensor fields are simpl y calle d tensor s i n th e literature . I n an y case , i t i s importan t t o understand clearl y what tensors are. Ou r purpose is to explain tensors to the readers as clearly as possible. Tenso r fields are defined b y means of tensors. The analytic theory o f tensor fields is known a s . Needles s to say, it is beyond the scope of linear algebra and w e shall not discus s it here. The notions of and pseudotensor ar e also important i n appli- cations of tensors. Sinc e they are related to representations o f linear groups, we shall mention onl y their definitions (§11.6) . As stated above , the notion o f tensor seems to have originated fro m wha t are no w calle d tenso r fields. Researc h ha s been done , sinc e the en d o f las t century, wit h th e stud y o f vecto r calculu s an d th e theor y o f invariants , o n systems o f function s whic h satisf y certai n transformatio n laws . Ricc i an d Levi-Civita founded tenso r calculus. I t became wel l known after bein g used by Einstein t o describ e th e theor y o f relativity . Whe n w e try t o describ e a physical phenomenon, w e use a . Thoug h th e descriptio n depends on the coordinat e system , the phenomenon itsel f doe s not. T o ex - plain the situation, systems of functions that satisfy a certain transformatio n law play a n importan t role . Dependin g o n th e typ e o f transformatio n law , several kinds of tensor () have been defined. B y the way, it seems that the word "tensor" was derived from tension . Thus , a tensor is classically define d as a syste m (o r a n array ) o f number s whic h satisfie s a certai n transforma - tion la w (se e §11.2) . W e can conside r that thi s syste m describe s an "object " whose existence does not depen d o n coordinate systems . Thi s point o f view

( )Se e R. Godement: Cour s c f algebre, Hermann, p. 269.

ix x PREFAC E is sufficient fo r computatio n but i t does not explain what the object is . In orde r t o defin e tensors , w e begin i n Chapte r I b y constructin g tenso r products o f vector spaces . Then , in Chapter II , we define tensor s and stud y their properties. I n many branches of mathematics the construction of tensor products is used as a powerful tool to construct new objects from known ones. For example, we will describe the extension o f a field of scalars (§I.8b) . Th e tensor product o f representations i s another example . In Chapter III, we discuss the notion o f exterior algebra. Exterio r algebra s are basic to the theory of differential forms . I n Chapter IV , to show anothe r aspect o f th e theor y o f tenso r products , w e discus s algebrai c system s wit h bilinear multiplication. I n particular, w e discuss Lie algebras. References fo r the English Edition

1. W . H. Greub, Multilinear algebra, Grundlehren Math . Wiss. , Bd. 136 , Springer-Verlag, Berlin and Ne w York, 1967 . In thi s book , tenso r algebra s ar e discusse d a t grea t length . Thi s i s th e second volume of Greub's text books on linear algebra; the first one is, Linear Algebra, 3rd ed., Grundlehren Math . Wiss. , Bd. 97 , Springer-Verlag, Berli n and New York, 1967 . In Bourbaki's series of Elements de Mathematique, is treated in: 2. N. Bourbaki, Algebre, Hermann, Paris, 1970 , Chapters 2 and 3. Though there are no books i n Japanese which are written abou t the sam e topics a s th e presen t volume , ther e ar e severa l book s o n linea r algebr a o r algebra, parts o f which are devoted to tensor algebras, e.g.: 3. I . Satake, Linear algebra, Marcel Dekker, New York, 1975 . The Japanes e editio n wa s publishe d b y Shokab o i n 1973 . A larg e par t of th e conten t o f th e presen t volum e i s explaine d neatl y i n Chapte r V . A description o f group representations i s also included. 4. N . Iwahori, Vector calculus, Shokabo, Tokyo, 1960 . (Japanese ) Tensors o f type (p, q) (p -f q < 2) o n R 3 ar e explained i n detail. Ther e are many example s and exercises . In som e books on differentia l geometry , tensor algebra an d it s relation t o geometry are explained a s preliminaries. 5. S . Sternberg , Lectures on , Prentic e Hall , Engle - wood Cliffs, NJ, 196 4 (republished b y Chelsea in 1983) . 6. S . Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, John Wile y & Sons, New York, 1963 . We consulted th e abov e books during th e preparation o f th e present vol - ume. The following books appeared mor e recently . 7. M . Marcus, Finite dimensional multilinear algebra, Parts I , II, Marce l Dekker, Ne w York, 197 3 (Part I ) and 197 5 (Part II).

127 128 REFERENCES FOR THE ENGLISH EDITION

8. L . Schwartz, Les tenseurs, Actualites scientifiques et industrielles, 1376 , Hermann, Paris, 1975 . 9. S . Lang, Algebra, 2nd ed., Addison-Wesley, Reading , MA, 1984 . 4. 3 contains only the beginning o f the theory o f Li e algebras. Fo r details, see e.g. 10. Y . Matsushima , Theory of Lie algebras, Kyoritsu Shuppan , Tokyo , 1956. (Japanese ) 11. N. Iwahori, Theory of Lie groups II, Iwanami Shoten , Tokyo, 1957 . 12. N. Jacobson, Lie Algebras, Interscience, New York, 196 2 (republishe d by Dover). 13. J . E . Humphreys, Introduction to Lie algebras and representation the- ory, 2nd printing, revised. Graduate d Texts in Math., vol. 9 , Springer-Verlag, New York, 1972 . Subject Inde x

Adjoint representation , 10 4 Derivations, 10 2 Algebra, 101 , 118 , 8 3 Algebraic closure o f a Lie algebra, 11 7 of A ®B, 1 7 Algebraic subgroup, 11 8 Dual , 2 Alternating (or skew-symmetric) tensor, 44 Dual space, 2 Alternating tensor space, 44, 77, 91 of , 4 9 Eddington's epsilon, 7 3 Alternator, 47, 77 Einstein's convention, 42 kernel of , 50 , 79 Engel's Theorem, 11 9 Annihilate, 99 Envelope, 94 Associated space , 95 Equivariant mapping , 6 9 Associative algebra, 55 , 103 Extension field , 2 0 center of , 6 6 Exterior algebra, 79 commutative, 6 6 Exterior product, 7 7 homomorphism, 56 , 79 set of generators for, 5 6 Factor algebra Associativity, 55 , 63, 78, 10 2 of algebra, 10 1 Axial vector, 7 2 of associative algebra, 5 6 of Li e algebra, 10 3 Bilinear forms, 3 , 34, 37, 72 Factor module, 3 0 Bilinear mapping , 3 Factor space, 26 Bilinear multiplication, 35 , 101 canonical mapping of , 2 6 Field of s Cartan's Theorem, 12 4 characterization o f extension of , 2 3 Characteristic, 1 extension of , 20 , 26 Characterization restriction of , 2 0 of exterior algebra, 8 2 Free associative algebra, 6 0 Chevalley, 105 , 108 , 11 7 Clifford algebra , 99 Generalized tensor, 7 1 Components, 3 6 components of , 7 1 Contraction, 4 1 Generalized tensor space, 71 Contravariant o f degree (or order) /?, 34 Graded Contravariant tensors , 34 associative algebra, 5 7 Co variant o f degree (o r order) #,3 4 ideal, 57 Covariant tensors , 34, 88 subalgebra, 5 7 alternating, 5 3 , 56 of degree 2, 34, 37 Grassmann algebra , 7 9 symmetric, 5 3 Group operating on the left, 6 8 Covariant vectors, 34 Group operating on the right, 68

Decomposable (elemen t o f A p(V)), 9 2 Homogeneous component, 5 7 Decomposable tensor, 5 5 Homogeneous elements, 5 7 Degree of the extension, 2 0 degree of, 5 7 Derivation, 6 1 Homogeneous ideal, 57

129 130 SUBJECT INDEX

Ideal Nilpotent generated b y S, 5 6 (linear) transformation, 105 , 119 of algebra, 10 1 Lie algebra, 11 5 of associative algebra, 5 6 , 10 5 of Lie algebra, 10 3 part, 105 , 109 Infinite direc t sum, 54 w-multilinear form, 1 1 Inner product, 8 8 ^-multilinear mapping , 1 1 Interior produc t (interio r multiplication) , Noncommutative polynomia l algebra, 6 0 90 Invariant subspace , 10 4 Orientation, 9 4 Invariant tensor , 10 6 /7-forms, 8 1 Jordan decomposition, 10 5 Plucker coordinates, 9 3 Plucker line, 93 Killing form, 12 3 Plucker relations, 9 6 Kronecker product, 1 6 Polynomial algebra, 67 Kronecker's symbol , 2, 40 Product, 36 , 54, 62 Projection, 4 6 Laplace expansion, 8 7 , 7 2 Left i?-modul e Pseudovector, 7 2 i?-basis of, 2 7 Pure (element of AP(V)), 9 2 free i?-module , 28 p-vectors, 81 set of generators of , 2 8 Lexicographical order, 8,1 5 Quadratic form, 9 8 Lie algebra, 10 2 Quotient rule , 42 algebraic, 11 7 center of , 103 , 121 /^-balanced mapping, 29 commutative, 11 5 /^-modules, 28 homomorphism, 10 3 Raising indices, 75 linear, 117 , 12 3 Rank nilpotent, 115 , 120 of an element o f AP(V), 9 5 of derivations, 10 3 Replica, 108 , 12 6 radical of , 117 , 12 1 of nilpotent transformation, 11 1 semisimple, 117 , 12 4 of semisimple transformation, 11 2 solvable, 115 , 122 Representation Lie subalgebra, 10 3 completely reducible, 10 5 Linear mapping , 3 contragradient, 10 4 extension of , 22 , 10 8 irreducible, 104 , 12 2 Linear transformations, 34 , 37, 43 reducible, 10 4 Linearize, 4 tensor product, 10 4 Lowering indices, 75 Representation (o f Li e algebra), 10 3 Representation space , 10 3 Matrix Right /^-module, 28 matrix unit, 1 6 , 27 of F{ ®F 2i 1 5 commutative, 2 8 of a linear mapping, 3 7 of linear mapping, 1 5 Scalars, 34 of the change of bases, 31, 38 Semisimple Minors, 84 (linear) transformation, 10 5 Module, 26 Lie algebra, 11 7 regarded a s Z-module, 28 matrix, 10 5 Module homomorphism, 3 0 part, 105 , 109 Multilinear form , 11 , 35 Signature, 43 alternating, 5 1 (5, i?)-bimodule, 30 symmetric, 5 1 Standard basis , 36, 79 Multilinear mapping, 1 1 ofA(V), 8 1 alternating, 5 1 Structure constants, 10 1 symmetric, 5 1 Subalgebra generated by 5, 5 6 SUBJECT INDEX 131

of algebra, 10 1 Tensors, 33 of associativ e algebra , 5 6 classical definitio n of , 4 0 Submodule, 3 0 of type (1,1), 34, 37 Symmetric algebra, 6 5 of type (1,2), 36, 10 1 characterization of , 6 7 of type («!,..., er), 33 , 44 of type (p,q), 3 4 Symmetric tensor space, 44, 62 product of , 4 0 dimension of , 4 8 Transformation la w of tensor components, Symmetrizer, 47, 62 39 kernel of , 49 , 64 o f linear mapping, 89

Tensor algebra, 5 5 Unit element , 27, 55 characterization of , 5 9 Universal enveloping algebra, 12 6 Tensor densities, 7 2 Universality (universa l property), 7 Tensor product of alternating tensor space, 52 associativity of , 1 3 of exterior algebra, 81 basis for, 8 of factor space , 26 canonical mapping of , 7 of symmetric algebra, 66 commutativity of , 1 0 of symmetric tensor space, 52 construction of , 6 , 7 of tensor algebra, 5 9 construction wit h generator s an d rela - of vector space generated b y a set, 25 tions, 25, 29 of n vector spaces, 1 2 Vector product, 7 2 of factor spaces , 32 Vector space, 28 of linear mappings, 14 , 19 generated by a set, 24 of matrices, 1 6 of bilinear mappings, 3 of modules , 30 of linear mappings, 2, 1 8 of subspaces, 1 0 of m x n matrices, 3 of vector spaces , 7 , 30 set of generators of , 5 properties (Tl) an d (T2 ) of , 4 subspace generated by 5, 5 property (T ) of , 5 subspace spanned by S, 5 Tensor space, 33 of type {p,q), 33 , 106 Copying an d reprinting . Individua l reader s o f thi s publica - tion, an d nonprofi t librarie s actin g fo r them , ar e permitted t o mak e fair us e o f the material, such as to cop y a chapter fo r use in teachin g or research . Permissio n i s granted t o quot e brie f passage s from thi s publication i n reviews , provide d th e customar y acknowledgmen t o f the source is given. Republication, systemati c copying , o r multipl e reproductio n o f any material i n this publication i s permitted onl y under license from the America n Mathematica l Society . Request s fo r suc h permissio n should b e addresse d t o th e Assistan t t o th e PubUsher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248. Request s ca n als o b e mad e b y e-mai l t o reprint - permissionQams. org. Recent Titles in This Series (Continued from the front of this publication)

73 Yu . G . Reshetnyak, Spac e mappings with bounded distortion , 198 9 72 A . V. Pogorelev, Bending s of surfaces an d stability of shells , 198 8 71 A . S. Markus, Introductio n to the spectral theory o f polynomial operator pencils, 198 8 70 N . I. Akhiezer, Lecture s on integral transforms, 198 8 69 V . N. Salii, Lattice s with unique complements, 198 8 68 A . G. Postnikov, Introductio n to analytic number theory, 198 8 67 A . G. Dragalin, Mathematica l intuitionism: Introductio n to proof theory, 198 8 66 Y e Yan-Qian, Theor y o f limit cycles, 198 6 65 V . M. Zolotarev, One-dimensiona l stabl e distributions, 198 6 64 M . M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-pose d problems of mathematical physic s and analysis, 198 6 63 Yu . M. Berezanskii, Sel f adjoint operator s in spaces of functions o f infinitely man y variables, 198 6 62 S . L. Krushkal', B. N. Apanasov, and N. A. Gusevskii, Kleinia n group s and uniformizatio n in examples and problems, 198 6 61 B . V. Shabat, Distributio n o f values of holomorphic mappings , 198 5 60 B . A. Kushner, Lecture s on constructive mathematical analysis , 198 4 59 G . P. Egorychev, Integra l representation and the computation o f combinatorial sums , 1984 58 L . A. Aizenberg and A. P. Yuzhakov, Integra l representations and residues in multidimensional comple x analysis, 198 3 57 V . N. Monakhov, Boundary-valu e problems with fre e boundaries fo r elliptic systems of equations, 198 3 56 L . A. Aizenberg and Sh. A . Dautov, Differentia l form s orthogona l to holomorphi c functions o r forms, and their properties, 198 3 55 B . L. Rozdesrvenskii and N. N. Janenko, System s of quasilinear equations and their applications to gas dynamics, 198 3 54 S . G. Krein, Ju. L Petunin, and E. M. Semenov, Interpolatio n o f linear operators, 198 2 53 N . N. Cencov, Statistica l decision rule s and optimal inference , 198 1 52 G . I. Eskin, Boundar y value problems fo r elliptic pseudodifferential equations , 198 1 51 M . M. Smirnov, Equation s of mixed type, 197 8 50 M . G. Krein and A. A. Nudel'man, Th e Markov momen t problem and extrema l problems, 197 7 49 I . M. Milin, Univalen t function s an d orthonormal systems , 197 7 48 Ju . V . Linnik and I. V. Ostrovskil, Decompositio n o f random variables and vectors, 1977 47 M . B. Nevel'son and R. Z. Has'minskii, Stochasti c approximation an d recursiv e estimation, 197 6 46 N . S. Kurpel', Projection-iterativ e method s fo r solutio n o f operator equations, 197 6 45 D . A. Suprunenko, Matri x groups, 197 6 44 L . I. Ronkin, Introductio n t o the theory of entire functions o f several variables, 197 4 43 Ju . L . Daleckii an d M. G . Krein, Stabilit y of solution s of differential equation s i n Banach space , 197 4 42 L . D. Kudrjavcev, Direc t an d invers e imbedding theorems, 197 4 41 I . C. Gohberg and I. A. Fel'dman, Convolutio n equation s and projection method s fo r their solution, 197 4 (See the AM S catalog for earlie r titles )