Tensor Space s and Exterio r Algebr a

Recent Titles in This Series

108 Take o Yokonuma, Tenso r spaces and exterior algebra , 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, and S. Shlosman, Wulf f construction : A global shape fro m

local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and thei r applications , 199 2 102 A . M. Il'in, Matchin g of 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 differentia l equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbit s in invariant theory , 199 2 99 Nori o Shimakura, Partia l differentia l operator s of elliptic type, 199 2 98 V . A. Vassiliev, Complement s o f discriminants of smooth maps : Topolog y an d

applications, 199 2 97 Itir o Tamura, Topolog y of foliations: A n introduction, 1 992 96 A . I. Markushevich, Introductio n t o the classical theory of Abelian functions , 199 2 95 Guangchan g Dong, Nonlinea r partia l differentia l equation s of second order, 199 1 94 Yu . S . Il'yashenko, Finitenes s theorems for limi t cycles , 199 1 93 A . T. Fomenko and A. A. Tuzhilin, Element s of the geometry and topology 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 Lie groups, I and II , 199 1 90 S . L. Sobolev, Som e applications of functiona l analysi s in mathematical physics , third

edition, 199 1 89 Valeri i Y. Kozlov and Dmitrii V. Treshchev, Billiards : A genetic introduction t o th e

dynamics of systems with impacts , 199 1 88 A . G. Khovanskii, Fewnomials , 199 1 87 Aleksand r Robertovich Kemer, Ideal s of identitie s of associative algebras, 199 1 86 V . M. Kadets and M. I. Kadets, Rearrangement s o f serie s in Banach spaces , 199 1 85 Miki o Ise and Masaru Takeuchi, Li e groups I, II, 199 1 84 Da o Trong Thi and 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 and S. G. Gindikin, Editors, Mathematica l problems of tomography, 199 0 80 Junjir o Noguchi and Takushiro Ochiai, Geometri c function theor y i n severa l 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 differentia l equations ,

1989 77 V . M. Filippov, Variationa l principle s for nonpotentia l operators , 198 9 76 Philli p A. Griffiths, Introductio n t o 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)

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 Mathematica l Societ y i? Providence , Rhod e Islan d

10.1090/mmono/108

T-vw&mttmft TENSORU KUUKA N TO GAISEKIDAISUU

(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 from the Japanese by Takeo Yokonuma

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

ABSTRACT. Thi s book provides an introduction to tensors and related topics. Th e book begins with definitions o f th e basic concepts of the theory; tensor products of vector spaces, tensors, tensor algebras, and exterior algebra. Their properties are then studied and applications given.

Algebraic systems with bilinear multiplication are introduced in the final chapter. I n partic-ular, the theory of replicas of Chevalley and several properties of Lie 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) Translation of: Tensor u kuukan to gaisekidaisuu. Includes bibliographical references and index. ISBN 0-8218-4564-0 1. Multilinear algebra. 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 by 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 T̂ X macro system. Visit the AMS 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 English Editio n vi i

Preface i x

Chapter I . Definition o f Tensor Product s 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 tenso r products o f mor e than tw o

vector spaces 1 1 §7. Tensor product s o f linear mapping s 1 4 §8. Examples of tensor product space s 1 8 §9. Construction o f tensor product s with generators an d relation s 2 4

§10. Tensor product s of i?-module 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 tenso r spaces 3 6 §3. Symmetric tensors and alternatin g tensors 4 3 §4. Tensor algebras and thei r properties 5 4 §5. Symmetric algebras and thei r propertie s 6 2 §6. Definition o f relativ e tensors (pseudotensor , tenso r density ) 6 8

Exercises 7 3

Chapter III . Exterior Algebra an d it s Applications 7 7 §1. Definition o f exterio r algebra and it s properties 7 7 §2. Applications t o determinants 8 3 §3. Inner (interior ) product s o f exterio r algebras 8 8 §4. Applications to geometry 9 1

Exercises 9 7

VI CONTENTS

Chapter IV . Algebraic Systems with Bilinea r Multiplication . Li e Algebras 10 1

§1. Algebraic systems with bilinear multiplication 10 1 §2. Replicas of matrice s 10 5 §3. Properties o f 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 i s a translation o f my work originally published by Iwanami Shoten , as a volume in their Lecture Serie s on linear algebra .

We assume, therefore, tha t reader s ar e familia r 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 thes e concept s i n th e text . W e have made som e changes in the references .

On thi s occasion , I woul d lik e t o expres s m y gratitud e t o Professo r Na -gayoshi Iwahori for giving me encouragement an d many valuable suggestions during th e preparatio n o f th e origina l wor k an d als o t o Mr . Hide o Ara i o f Iwanami Shote n fo r continue d support .

I am grateful t o the American Mathematica l 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 Yokonum a fo r thei r excellen t typing .

Takeo Yokonum a December 199 1

This page intentionally left blank

Preface

The subject matte r of the present book is generally called multilinear alge-bra; we shall discus s mainl y tensor s an d relate d concepts . Th e reader s 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 i s 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 tenso r 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 i s to explain tensors to the readers as clearly a s possible. Tenso r fields are defined b y means of tensors . The analytic theory o f tensor fields is known a s tensor calculus . Needles s t o say, it i s beyond th e scope of linear algebra and w e shall not discus s i t here . The notions of tensor densit y an d 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 th e 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 founde d tenso r calculus . I t became wel l known afte r 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 coordinat e system . Thoug h th e descriptio n depends o n th e coordinat e system , th e phenomenon itsel f doe s not . T o ex -plain the situation, systems of functions tha t 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 (field) 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 (see §11.2) . W e can conside r tha t thi s syste m describe s a n "object " whose existence does not depen d o n coordinate systems . Thi s point o f view

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

ix

x PREFAC E

is sufficient fo r computatio n bu t i t does not explain what the objec t 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 vecto r spaces . Then , i n Chapte r II , we define tensor s an d 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 know n ones. For example, we will describe the extension o f a field of scalar s (§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 differentia l 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 , we discuss Lie algebras.

References for the English Edition

1. W. H. Greub, Multilinear algebra, Grundlehre n Math . Wiss. , Bd. 136 , Springer-Verlag, Berlin an d New 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. , Grundlehre n Math . Wiss. , Bd. 97 , Springer-Verlag, Berli n and New York, 1967 .

In Bourbaki's series of Elements de Mathematique, tensor algebra is treated in:

2. N . Bourbaki, Algebre, Hermann, Paris , 1970 , Chapters 2 and 3 . Though there ar e no books in Japanese which ar e 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 ar e 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 of 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 differential geometry, Prentic e Hall , Engle -wood Cliffs , NJ , 196 4 (republishe d b y Chelsea in 1983) .

6. S . Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, John Wiley & Sons, New York, 1963 .

We consulted th e abov e books during th e preparation o f th e presen t vol -ume.

The following books appeare d mor e recently .

7. M . Marcus , Finite dimensional multilinear algebra, Parts I , II , Marce l Dekker, New York, 197 3 (Par t I ) and 197 5 (Par t II) .

127

128 REFERENCES FOR THE ENGLISH EDITION

8. L . Schwartz, Les tenseurs, Actualites scientifiques e t industrielles, 1376 , Hermann, Paris , 1975 .

9. S . Lang, Algebra, 2nd ed. , Addison-Wesley, Reading , MA, 1984 . Section 4. 3 contains only the beginning o f the theory o f Lie 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 Algebra, 101 , 118 Algebraic closure of a Lie algebra, 11 7 Algebraic subgroup, 11 8 Alternating (or skew-symmetric) tensor, 44 Alternating tensor space , 44, 77, 91

dimension of , 4 9 Alternator, 47 , 77

kernel of , 50 , 79 Annihilate, 99 Associated space , 95 Associative algebra, 55 , 103

center of , 6 6 commutative, 66 homomorphism, 56 , 79 set of generators for , 5 6

Associativity, 55 , 63, 78, 102 Axial vector, 72

Bilinear forms , 3 , 34, 37, 72 Bilinear mapping , 3 Bilinear multiplication, 35 , 101

Cartan's Theorem, 12 4 Characteristic, 1 Characterization

of exterior algebra, 82 Chevalley, 105 , 108 , 117 Clifford algebra , 99 Components, 36 Contraction, 41 Contravariant o f degree (or order) /?, 34 Contravariant tensors , 34 Co variant o f degree (o r order) #,3 4 Covariant tensors , 34 , 88

alternating, 5 3 of degree 2 , 34 , 37 symmetric, 5 3

Covariant vectors , 34

Decomposable (elemen t o f A p(V)), 9 2 Decomposable tensor , 55 Degree of the extension, 20 Derivation, 61

Derivations, 10 2 Determinant, 8 3

of A ®B, 1 7 Dual basis, 2 Dual space , 2

Eddington's epsilon , 73 Einstein's convention, 42 Engel's Theorem, 11 9 Envelope, 94 Equivariant mapping , 69 Extension field , 2 0 Exterior algebra, 79 Exterior product, 7 7

Factor algebra of algebra, 10 1 of associative algebra , 56 of Lie algebra, 10 3

Factor module , 30 Factor space, 26

canonical mapping of, 2 6 Field of scalar s

characterization o f extension of , 2 3 extension of , 20 , 26 restriction of , 2 0

Free associative algebra , 60

Generalized tensor , 71 components of , 7 1

Generalized tensor space , 71 Graded

associative algebra , 57 ideal, 57 subalgebra, 57 vector space, 56

Grassmann algebra , 79 Group operating on the left , 6 8 Group operating on the right , 68

Homogeneous component, 5 7 Homogeneous elements , 57

degree of, 5 7 Homogeneous ideal , 57

129

130 SUBJECT INDEX

Ideal generated b y S, 56 of algebra , 10 1 of associative algebra, 56 of Lie algebra, 10 3

Infinite direc t sum, 54 Inner product , 8 8 Interior produc t (interio r multiplication) ,

90 Invariant subspace , 10 4 Invariant tensor , 10 6

Jordan decomposition , 10 5

Killing form, 12 3 Kronecker product, 1 6 Kronecker's symbol , 2, 40

Laplace expansion , 87 Left i?-modul e

i?-basis of, 2 7 free i?-module , 28 set of generators of, 2 8

Lexicographical order , 8,1 5 Lie algebra, 10 2

algebraic, 11 7 center of , 103 , 121 commutative, 11 5 homomorphism, 10 3 linear, 117 , 123 nilpotent, 115 , 120 of derivations , 10 3 radical of , 117 , 121 semisimple, 117 , 124 solvable, 115 , 122

Lie subalgebra, 10 3 Linear mapping , 3

extension of , 22 , 108 Linear transformations, 34 , 37, 43 Linearize, 4 Lowering indices, 75

Matrix matrix unit , 1 6 of F{ ®F 2i 1 5 of a linear mapping, 37 of linear mapping, 1 5 of the change of bases, 31, 38

Minors, 84 Module, 26

regarded a s Z-module, 28 Module homomorphism, 3 0 Multilinear form , 11 , 35

alternating, 5 1 symmetric, 51

Multilinear mapping , 1 1 alternating, 5 1 symmetric, 51

Nilpotent (linear) transformation , 105 , 119 Lie algebra, 11 5 matrix, 10 5 part, 105 , 109

w-multilinear form, 1 1 ^-multilinear mapping , 1 1 Noncommutative polynomia l algebra , 60

Orientation, 9 4

/7-forms, 81 Plucker coordinates, 93 Plucker line, 93 Plucker relations, 96 Polynomial algebra , 67 Product, 36 , 54, 62 Projection, 4 6 Pseudotensors, 72 Pseudovector, 7 2 Pure (element o f AP(V)), 9 2 p-vectors, 81

Quadratic form, 9 8 Quotient rule , 42

/^-balanced mapping , 29 /^-modules, 28 Raising indices, 75 Rank

of an element o f AP(V), 95 Replica, 108 , 126

of nilpoten t transformation , 11 1 of semisimple transformation, 11 2

Representation completely reducible , 105 contragradient, 10 4 irreducible, 104 , 122 reducible, 10 4 tensor product , 10 4

Representation (o f Lie algebra), 10 3 Representation space , 10 3 Right /^-module, 28 Ring, 27

commutative, 28

Scalars, 34 Semisimple

(linear) transformation, 10 5 Lie algebra, 11 7 matrix, 10 5 part, 105 , 109

Signature, 43 (5, i?)-bimodule, 30 Standard basis , 36 , 79

ofA(V), 8 1 Structure constants , 10 1 Subalgebra

generated by 5, 5 6

SUBJECT INDEX 131

of algebra , 10 1 of associativ e algebra , 56

Submodule, 30 Symmetric algebra , 65

characterization of , 6 7 Symmetric tensor , 44 Symmetric tensor space , 44, 62

dimension of , 4 8 Symmetrizer, 47 , 62

kernel of , 49 , 64

Tensor algebra, 5 5 characterization of , 5 9

Tensor densities , 72 Tensor produc t

associativity of , 1 3 basis for, 8 canonical mappin g of , 7 commutativity of , 1 0 construction of , 6 , 7 construction wit h generator s an d rela -

tions, 25 , 29 of n vector spaces , 12 of factor spaces , 32 of linear mappings , 14 , 19 of matrices , 16 of modules , 30 of subspaces , 1 0 of vector spaces , 7 , 30 properties (Tl ) an d (T2 ) of, 4 property (T ) of , 5

Tensor space, 33 of type {p,q), 33 , 106

Tensors, 33 classical definition of , 4 0 of type (1,1) , 34 , 37 of type (1,2), 36, 101 of type («!,. . . , er), 33 of type (p,q), 3 4 product of , 4 0

Transformation la w of tensor components, 39

Transpose of linear mapping, 89

Unit element , 27 , 55 Universal enveloping algebra, 12 6 Universality (universa l property), 7

of alternating tensor space , 52 of exterior algebra , 81 of factor space , 26 of symmetric algebra, 66 of symmetric tensor space , 52 of tensor algebra, 59 of vector space generated by a set, 25

Vector product, 7 2 Vector space, 28

generated by a set, 24 of bilinear mappings, 3 of linear mappings, 2 , 18 of m x n matrices, 3 set of generators of, 5 subspace generated by 5, 5 subspace spanned by S, 5

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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 t o 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 t o analytic number theory, 198 8 67 A . G. Dragalin, Mathematica l intuitionism: Introductio n t o proof theory , 198 8 66 Y e Yan-Qian, Theor y of 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 physics 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 groups 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 mathematica l analysis , 198 4 59 G . P. Egorychev, Integra l representation an d the computation o f combinatoria l 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 boundarie s fo r ellipti c 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 thei r

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 severa l variables, 197 4 43 Ju . L . Daleckii and M. G. Krein, Stabilit y of solution s of differentia l equation s in

Banach space , 197 4 42 L . D. Kudrjavcev, Direc t an d inverse 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 AMS catalog for earlie r titles )