Convergence Structures and Applications II: Proceedings of the conference held at Schwerin, GDR from May 16 to 20, 1983 [Reprint 2021 ed.] 9783112528006

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ABHANDLUNGEN DER AKADEMIE DER WISSENSCHAFTEN DER D D R Abteilung Mathematik, Naturwissenschaften, Technik

N2

1984

Convergence Structures and Applications, II

Akademie-Verlag • Berlin

Iffi*)

ABHANDLUNGEN DER A K A D E M I E DER WISSENSCHAFTEN DER D D R Abteilung Mathematik — Naturwissenschaften — Technik Jahrgang 1984 • Nr. 2 N

Convergence Structures and Applications II Proceedings of the Conference held at Schwerin, G D R from May 16 to 20, 1983

Herausgegeben von

S. Gähler, W. Gähler und G. Kneis

AKADEMIE-VERLAG

BERLIN • 1984

Herausgegeben im Auftrag des Präsidenten der Akademie der Wissenschaften der D D R von Vizepräsident Prof. Dr. Heinrich Scheel

Dieser Titel wurde vom Originalmanuskript des Autors reproduziert

ISSN 0138-1059

Erschienen im Akademie-Verlag, DDR-1086 Berlin, Leipziger Straße 3—4 © Akademie-Verlag Berlin 1984 Lizenznummer: 202 • 100/069/84 Printed in the German Democratic Republic Gesamtherstellung: VEB Kongreß- u. Werbedruck, 9273 Oberlungwitz LSV 1035 Bestellnummer: 763 336 6 (2001/84/2 N) 03800

PREFACE The Mathematical Society of the GDR and the Institute of Mathematics of the A c a d e m y of Sciences of the GDR organized the conference "Convergence Structures and Applications,

II"

from May 16 to 20, 1983 at Schwerin. The first such conference in GDR was held 1978 at Frankfurt/Oder

1)

.

There are two important developments of the theory of convergence structures In the last time. On one hand the topic of applications of proper convergence structures is growing permanently. On the other hand the notion of convergence structure has been generalized in such a manner that the theory of convergence structures becomes more and more a general structure theory. According to this development structure theoretical investigations came in the field of interest. Participating at the Schwerin conference were the following 32 mathematicians : Averbuch, V. I. (Moscow)

Kratochvil, P. (Prague)

Beattie, R. (Sackville)

Lindström, M . (Xbo)

Boseck, H. (Greifswald)

Lowen, R. (Brussels)

Butzmann, H.-P. (Mannhelm)

Lowen-Colebunders, E. (Brussels)

Czichowski, G. (Greifswald)

Porst, H.-E.

Firmanty, M. (Szczecin)

Prleß, W . (Berlin)

Frig, R. (Kosice)

Przybylski, B. (fco'dz)

(Bremen)

Gähler, S. (Berlin)

Rudolph, K.-P.

Gähler, W . (Berlin)

Sasin, W . (Warsaw)

Haase, H. (Greifswald)

Schmidt, H.-J.

Herrlich, H . (Bremen)

Spakowski, A.

(Greifswald) (Potsdam-Babelsberg) (Szczecin)

Hoehnke, H.-J. (Berlin)

Sulk, I. (Greifswald)

Ivanov, A . A . (Leningrad)

Terpe, F.

Janeczko,- S. (Warsaw)

(Greifswald) o Vainio, R. (Abo)

Kneis, G. (Berlin)

Zekanowski, Z. (Warsaw)

Koutnik, V. (Prague)

Zieminska, J.

(Szczecin)

In Schwerin 30 talks were presented of length between half an hour and one h o u r . Th.ls volume consists of the proceedings of the

conference

and of additionally submitted papers (with starred titles in the table of contents) . "1) The proceedings of this conference were published in a separate volume titled "Convergence Structures and Applications to Analysis" in "Abhandlungen der Akademie der Wissenschaften der DDR", 1979, Nr. 4N.

The contributions of this volume may be divided according to the fol lowing topics: 1° Proper convergence structures, especially filter convergence and »sequential structures; Cauchy structures (H.-P. Butzmann, R. Fric/V. Koutnik, R. Fric/F. Zanolin, D. C. Kent/G. D . Richardson, G. Kneis, P. Kratochvíl,

Lowen-Cole-

bunders, P. Mikusinski/J. Pochcial, E. Pap, J. Pochcial, H.-J. Schmidt, R. Vainio) 2° General theory of convergence structures and categorical

topology

a n d algebra, fuzzy convergence (W. Gáhler, H. Herrlich, H.-J. Hoehnke, A . A. Ivanov, R. Lowen, H . - ¿ . Porst) 3° Applications of convergence structures, in particular to functional analysis, non-linear analysis, numerical mathematics, measure theory, summation theory, extremal theory (V. I. Averbuch, R . Beattie, S. Gáhler, H . Haase, M . P. Kaz, A . U. Khrennikov, M. LindstrSm. A. Spakowski, J. Sulk, F. Terpe) 4° Differential calculus and structures in differential

geometry,

especially generalized differential structures", manifolds of mappings, general Lie theory (H. Boseck, M. Firmanty, 'II. Gáhler, 3. Janeczko, A . Kriegl, P. W. Michor, Z. Pasternak-Winiarski,

B. Przybylskl, '.V. Sasin/

Z. Zekanowski, R. Schmid, Z. Zekanowski).

Berlin, July 11, 1983

4

The editors

CONTENTS Averbuch, V. I., The Asplund-Rockafellar-Gregory Theorem and. Pseudo-Topologies

7

Beattie, R., The Closed Graph Theorem in Convergence Spaces with Webs

9

Boseck, H., Graded lie Algebras, Graded Hopf Algebras, and Graded Algebraic Groups

15

Butzmann, H.-P., Some Remarks on Extremal Compactifications of Convergence Spaces

25

Firmanty, M., Constructions of Generalized Differentiable Manifolds by Means of Path Structures

31

Fric, R., and V. Koutnik, Recent Development in Sequential Convergence

37

Fric, R., and P. Zanolin, A Sequential Convergence Group Having No Completion

47

Gahler, 3., Contributions.to the Theory of Discrete Convergence

49

Gahler, W., A General Theory of Convergence Structures

57

Gahler, W., On Generalizations of Differentiable Manifolds

65

Haase, W., On Convergence in Distribution of Closed Random Sets

73

Herrlich, H., Topological Structure Theory

77

Hoehnke, H.-J., Subfunctors Associated with Quasivarieties

81

Ivanov, A. A., Topological Type Structures

85

Janeczko, 3., On G-Stable Lagrangian Submanifolds

91

Kaz, M. P., On Some Properties of Linear Pseudo-Topologies Kent, D. C., and G. D. Richardson, Cauchy Spaces and Their Completions *

97 103

Khrennikov, A. U., Some Applications of Keller's Convergence Structure to the Nonlinear Functional Analysis*

113

Kneis, G., Completion for Some Concrete Categories over the Category of Uniform limit Spaces Koutnik, V., see R. Frio

119 37

Kratochvil, P., On the Form of Neighborhoods in Closure Spaces

131

Kriegl, A., AConvex Convenient Setting of Differential Calculus in Locally Spaces*

135

Lindstrom, M., Precompact Mappings between Equable Convergence Vector Spaces*

141

Lowen, R., Fuzzy Convergence versus Weak Convergence in Spaces of Probability Measures

147

Lowen-Colebunders, E., An Application of Cauchy Spaces to the Theory of Function Classes

153

5

Michor, P. W., Applications of Hamilton's Inverse Function Theorem to Manifolds of Mappings *

159

Mlkuslnskl, P., and J. Poohclal, On Bases of Convergence *

163

Pap, £., Triangular Funotlonals on Commutative Semigroups*

167

Pasternak-Wlnlarskl, Z., Differential Groups of Class

173

Dq *

Poohcial, J., On Metrlzability of Convergence *

177

Pochclal, J., see P. Mlkuslnskl

163

Porst, H.-E., Regular Decompositions of Semltopologlcal Functors

181

Przybylskl, B., Characterizations of Differential Spaces by Topological Algebras

189

Richardson, G. D., see D. C. Kent

103

Sasln, W., and Z. Zekanowskl, On Stokes" Theorem In Differential Spaces

195

Schmid, R., The Inverse Function Theorem of Nash and Moser for the

r-Differentiability *

201

Schmidt, H.-J., Locally Non-Euclidean Topologies in Space-Time .

207

Spakowski, A., Convergences and Topologies for Extremum Problems

213

'Sulk, I., Uniform Convergence in the Space of Summations

217

Terpe, F., Moore-Smith-Sequences of Measurable Functions

223

Vainio, R., On Connectedness in Limit Space Theory

227

Zanolin, F., see R. Fric

47

Zekanowskl, Z., On Quasl-2-Metrics and Pseudo-Riemannian Metrics Generated by Them on a Differentlable Manifold

233

Zekanowskl, Z., see '.V. Sasin

195

6

:

THE ASPLUND-ROCKAFELLAR-GREGORY THEOREM AND PSEUDO-TOPOLOGIES V. I. Averbuch 0. Intro duo til on The aim of this paper is to generalize, using pseudo-topologies, a result of Gregory £6 J ooncerning the equivalence of the directional differentiability of a continuous convex function and the upper semicontinuity of its subdifferential. This result is in turn a generalization of a result of Asplund and Rockafellar ClJ to the case where the subdifferential is not a singleton. The Asplund-Rockafellar-Gregory theorem covers only the case of differentibilities, associated with the topologies of uniform convergence on set systems. Our generalization allows to cover the case of differentiabilities, associated with the pseudo-topologies of convergence on filter systems, in particular the MB- and FB-differentiabilitiea [4]. 1. Notation and definitions Let X be a separated locally convex topological veotor space (over the reals). We denote by C and IP the filters of neighborhoods of 0 in X and E, respectively. Let f be a real-valued continuous convex function on X. For x £ X, t > 0 and h 6 X we define f t (h):^ f ( x+t i^- f ( x ), df (x,h):=lim f^.(h), and 0

i cf.

Then f is directionally

"Ej-dlfferentiable at x iff its subdifferential mapping ?f is T y -upper semi-continuous at x. If ¿7 is the filter system generated in a natural sense by a system Jr of bounded sets in X such that A f c i / 4 , A > 0 ^ X k e J 9 - , then«? satisfies the conditions of the theorem. In this special case we obtain the Asplund-Rockafellar-Gregory theorem (tsZ, Theorem 3.1). Other examples of suitable systems e? are MB and FB. In this case we obtain characterizations of the directional MB- and FB-differentiability of continuous convex functions. Corollary. For each continuous convex function f on X its subdifferential mapping is T ^ - u p p e r semi-continuous (at every point). This follows from the fact that every continuous convex function on X is everywhere directionally MB-differentiable (cf.

)-

For the equable (lukovichny) modifications Vjf ([5], E2J) of pseudotopologies Tj> the theorem seems to fail (cf. ¡143, App. 5, A). But the following result holds (cf. C4j): Proposition. Let X and f be as above. Then, if 3 f is "T^-upper semi-continuous at x, then f is directionally HL*-differentiable at x. # Here

:=

# =

The definition of directional HL 1 -differen-

tiability is quite analogical to the definition of HL'-differentiability

(CO).

References Cll

Asplund, E., and R. T. Rockefeller, Gradients of convex functions, Trans. A. M. S. 139 (1969), 443-467. ["21 ABepGyx, B. Li., 0 HenpepiiBHOCTi: K0!.Ti0Hi;p0BUHL4,i, Math. Kachr. 75 (1976), 153-183. [•3] A B e p d y x , 3 . L i . , BanyKJiaa •jiyHKiuiH, si^wepeHuwpyeuaii no I ' a i o , peHUHpyeMas no iviai'lKJiy-EacTMaHK, Proc. Intern. Summer School on Convergence Structures and Appl. to Anal. (Frankfurt/O. 1978), Abh. Akad. Wiss. DDR, Nr. 4£ (1979), 7-9. C4J

ABepfiyx, 3. Li., »1 0 . Eyxepa £ 5 ] .

m

i p e j i n x e p , A . , H 3 . E y x e p , Miuj^epeKuiiajiLHoe ticmicjieiiKe 3 BCKTODHEX n p o c i p a H C T s a x 6e3 HOPMLI, j l u p , ;,iocxBa 1 9 7 0 .

C6J

8

R. CMOJIHHO:-», /lonojinemin K Km:re ¿»piijmxepa K

Gregory, D. A., Upper semi-continuity of subdifferential mappings, Canad. Math. Bull. 23 (1980), 11-19.

THE CLOSED GRAPH THEOREM IN CONVERGENCE SPACES WITH WEBS Ronald Beattie

1) '

1) Introduction The first closed graph theorem, due to Banach [1], stated that if E and F are Frechet spaces, i.e. complete metrizable locally convex topological vector spaces and f : E »F is a linear mapping with a closed graph, then f is continuous. This theorem has proved to be one of the most important in analysis and there have been many attempts to generalize it to locally convex topological vector spaces (lcs). Among the most successful were those of Ptak [12] and Kelly [ 9 ] . Using different methods they showed the following: if E is an arbitrary lcs, F is a fully complete lcs and f : E >F is a linear mapping which has a closed graph and is nearly continuous, then f is continous. We explain these terms briefly. The notion of power convergence, i.e. convergence of a filter to a set has been defined in [9] and generalized in [3]. With respect to this notion, a linear mapping f : E »F is nearly continuous if, whenever a filter F converges in E , the image filter is power Cauchy in F , i.e. it is trying to converge to a set. Full completeness is the property that every equable or scalar [9] power Cauchy filter power converges. Dually, f is nearly continuous if its adjoint mapping between the continuous dual spaces f* : /. (F) * /-C(E) is continuous on its domain of definition. Full completeness is then the property that closed subspaces of 1-£(F) are weak*-closed [2], While the closed graph theorem was a strong generalization of Banach's, it had a number of difficulties. It is, in general, difficult to tell when a mapping is nearly continuous except in special cases, for example when E is barrelled [13]. But the main problem is that the class of fully complete spaces is small and difficult to work with. It has few permanence properties and excludes some of the more interesting spaces for analysis, e.g. the distribution spaces. Grothendieck [8] conjectured that by restricting attention to Frechet spaces E , the class A of spaces F for which a closed graph theorem would hold for a mapping f : E »F would have good permanence properties. In addition, a closed graph theorem for Frechet spaces would easily give rise to one for ultrabornological spaces so that if the class A were large enough, the closed graph theorem would encompass those lcs of importance in analysis. M. De Wilde [6] has provided a positive answer to this conjecture with his class of spaces with webs. ^ Dept. of Mathematics, Mount Allison University, Sackville, N.B., EOA 3C0, Canada

9

In this p a p e r we look at De W i l d e ' s webs from the point of view of c o n v e r g e n c e spaces a n d relate his theory to the p r e v i o u s w o r k of Ptak and K e l l y .

2)

Web-spaces Let

P

be a v e c t o r space. A web

b a l a n c e d subsets of

F

consists of a sequence W^

on

F

is a countable family

. The c o l l e c t i o n

W^j

, as b o t h

i

and

j

constitutes the second layer. In the same w a y , for e a c h set

of

layer

(W.) w h o s e u n i o n a b s o r b s F . For e a c h set 1 ) o f subsets of ^ W^ w h o s e u n i o n , as j

there is a sequence varies, absorbs

W

a r r a n g e d in layers as follows. The first

W. 1

vary,

W..

there

is a sequence

(W. .. ) of subsets of W. . w h o s e u n i o n , as k varies, d 1J 1J K as a b s o r b s e a c h " p o i n t of W.y . The c o l l e c t i o n i > J and k vary c o n s t i t u t e s the third layer. And so on. By a strand

S

W., W.., W....... l 1J lj k

of the web

W , we m e a n any sequence of sets

one from e a c h layer, e a c h set, from the second o n ,

b e i n g one of those d e t e r m i n e d by its p r e d e c e s s o r . E a c h strand is clearly a filter base and the c o l l e c t i o n of strands g e n e r a t e s a c o n v e r g e n c e space in the sense of Kent

[10]. T h e r e is no r e a s o n to believe t h a t , in

g e n e r a l , this w i l l be a c o n v e r g e n c e space in the stronger sense tierung

the vector space o p e r a t i o n s . W i t h this in m i n d we say that v e c t o r web on strand

(Limi-

[7]) or that the c o n v e r g e n c e structure w i l l be c o m p a t i b l e

S

F , if, for any two strands S'

such that

+ S''

S' , S''

W

with

is a

there is a n o t h e r

=> S. W h i l e it is not clear how

restric-

tive this c o n d i t i o n is, it w i l l be shown to be s a t i s f i e d by a large class of spaces. H e n c e f o r t h If

til

is a web on

by the strands is called the d e n o t e d by ping

"web"

web-space

F ^ . We say that a cvs

i : F^

filter base in

»F

will mean

"vector web" .

F , the c o n v e r g e n c e v e c t o r space F

(cvs)

associated with

W

generated and is

has a-web if the identity

F . F i n a l l y , we say

F

(i)

F^

is

(ii)

F^

is fully

if, in

hold:

complete complete

T h a t c o m p l e t e n e s s and full c o m p l e t e n e s s c o m p l e t e n e s s conditions)

til

has a strict web

a d d i t i o n , one of the f o l l o w i n g equivalent c o n d i t i o n s

coincide on

F^

(as well as several

It should be n o t e d that w e b - s p a c e s have a very strong p r o p e r t y , namely: there is a countable family a basis from

10

III

[5] lcs.

countability

of sets such that for

F , there is a coarser convergent filter

III . This c o n d i t i o n is m u c h stronger than first

bility for a cvs.

other

is a s u r p r i s i n g result

s i n c e , in g e n e r a l , these p r o p e r t i e s are very d i f f e r e n t e v e n for

every n u l l filter

map-

is c o n t i n u o u s , i.e. if e a c h s t r a n d is a convergent

5

having

counta-

3) Examples and permanence properties Example 1. Let basis of

be a Frechet space and let (U ) be a neighbourhood n 1 such that U , „ c U for all n . For the first layer n+1 £ n VT = iU 1 for all i . For the second layer, W ^ = iU 2 for

0

ill , let

of all

F

i, j . For the third layer W... = iU, ij K J ill

Then

is a strict web for

The web-space

F^ = F .

Example 2 . Let

L^(F)

U? 1

and has essentially only one 'strand.

W , let

for all

W^ = U°

for all

i . For the second

i,j . For the third layer, W... = - 4 U? • XJ K

ill

Continue in this manner.

L (F) and has counta- • c i s ^he Mackey or bornological

is a strict web for

bly many strands. The web space convergence on

i,j,k. And so on.

be the continuous dual of the Frechet space above.

For the first layer of layer, W.. = 1J

F

for all

('-j.

)(t/

i-Q(F) .

The above two examples in conjunction with the following theorem generate a huge class of cvs having strict vector webs, among them, the spaces of test functions and distributions. Theorem (1) If a cvs on

F

F

has a strict web

such that

(2) If a cvs

F^

has a strict web

F

then there is a strict web (3) If a cvs

F

W

» F^ , then

W^

on

has a strict web

W ill M W

linear surjection onto another cvs such that

G^,

is a quotient of

and

X

and

M

M^

f : F

G , then under

F^ .

is a closed subspace of

such that and

F^

is any other cvs structure

is a strict web for

G

F,

= F^IM .

»G

is a continuous

has a strict web

W

f .

In particular, quotients of cvs with strict webs have strict webs. (4) If

(F n )

product

i-s

] F n6N n

(5) If

W

a

is

P

(6) If

= P r °J

ill n

such that

V w

has a strict web

Wp

=

ind

F = ind F

^ V »

n

, then the

= I • | (F ).. n n€N % Wn

then the

such that

•

n

(F ) is a sequence of cvs with strict webs

I

Wn

F.. w n

sequence of cvs with strict webs F = proj F^

(

inductive limit X

sequence of cvs with strict webs

has a strict web

projective limit P

a

has a strict web

Wn

, then the

ill^ such that

•

11

is

(7) If d i r e c t sum

a

sequence of cvs w i t h strict w e b s

2— F nEN n

has a strict web

=n€N2Z

W_

Wn

, t h e n the

such that

E

(F„)w " n

The c o n s t r u c t i o n of the w e b s for e a c h of the above p a r t s can be in

[6], [11], or [14]

found

. T h a t the c o r r e s p o n d i n g w e b - s p a c e s behave as

i n d i c a t e d in the t h e o r e m is, in e a c h case, a s t r a i g h t f o r w a r d

verifica-

tion. The above t h e o r e m shows, in p a r t i c u l a r , that the class of w e b spaces t h e m s e l v e s has e x c e l l e n t p e r m a n e n c e

4) The c l o s e d g r a p h

properties.

theorem

We now give the c l o s e d g r a p h t h e o r e m for cvs w i t h w e b s : T h e o r e m . Let f : E

»F

E

be a F r e c h e t s p a c e ,

F

a cvs w i t h strict web

a linear m a p p i n g w i t h a closed g r a p h . T h e n

is c o n t i n u o u s a n d c o n s e q u e n t l y also

f : E

E , then

is n e a r l y c o n t i n u o u s Since

W

is strict,

v e r g e s in in

Fw

(f(U ))

)

is power Cauchy on

is a n e i g h b o u r h o o d F^

(see [4] or [14]). In a d d i t i o n F^

is fully c o m p l e t e a n d so

. Since the g r a p h of

f

E * F w and it easily follows that

and

»F .

P r o o f . We shall only outline the p r o o f . If basis for

W

f : E——»F^

is closed in

i.e.

f : E

»F^

(f(U )) is scalar. (f(Un)) power E*F

, it is

(f(U )) c o n v e r g e s to

0

conclosed

in

The f o l l o w i n g is a n easy g e n e r a l i z a t i o n of the a b o v e t h e o r e m

F^ . (see

(see [4]) : T h e o r e m . Let

E

be a cvs

a cvs w i t h a strict web closed graph. Then

inductive limit of F r e c h e t s p a c e s , let W

f : E

and »F^

f : E

>F

F

be

a linear m a p p i n g w i t h a

is c o n t i n u o u s and so also

f : E

>F .

The point of the above t h e o r e m is not a g e n e r a l i z a t i o n of De W i l d e ' s c l o s e d g r a p h t h e o r e m from lcs to cvs but r a t h e r an i l l u s t r a t i o n of how c o n v e r g e n c e spaces can be u s e d in f u n c t i o n a l a n a l y s i s . Not only do the w e b - s p a c e s allow a m u c h sharper f o r m u l a t i o n of the c l o s e d g r a p h (the c o n t i n u i t y is into

F^

a n d not just

theorem

F ) but they also relate

the

theory of w e b s to the e a r l i e r theory of fully complete spaces of Kelly a n d Ptak.

12

References

[1] B a n a c h , S . , T h é o r i e des o p é r a t i o n s l i n é a i r e s , W a r s a w

1932.

[2] B e a t t i e , R . , C o n t i n u o u s c o n v e r g e n c e and the c l o s e d g r a p h t h e o r e m , M a t h . N a c h r . 99 (1980), [3]

87-94.

B e a t t i e , R . , H y p e r c o m p l e t e c o n v e r g e n c e s p a c e s , to a p p e a r Math.

[4] B e a t t i e , R . , C o n v e r g e n c e Math.

in

Nachr. spaces w i t h w e b s , to a p p e a r

in

Nachr.

[5] B e a t t i e , R . , C o m p l e t e n e s s in web s p a c e s ,

preprint.

[6] De W i l d e , M . , C l o s e d g r a p h theorems and w e b b e d

spaces,

R e s e a r c h N o t e s in M a t h e m a t i c s 19, P i t m a n , L o n d o n [7] F i s c h e r , H . R . , L i m e s r ä u m e , M a t h . Ann. 137

(1959),

1978. 169-303.

[8] G r o t h e n d i e c k , A . , P r o d u i t s t e n s o r i e l s t o p o l o g i q u e s et

espaces

n u c l é a i r e s , Mem. A m e r . M a t h . Soc. 16, 1955 [9]

K e l l e y , J . L . , H y p e r c o m p l e t e linear t o p o l o g i c a l M i c h i g a n M a t h . J. 5 (1958),

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13

GRADED LIE ALGEBRAS, GRADED HOPF ALGEBRAS, AND GRADED ALGEBRAIC GROUPS Helmut Boseck ^ ^ The development of graded Lie algebras was essentially influenced by considerations on supersymmetry in modern physics. In mathematics a differential geometric treatment of graded Lie theory was given by B.KONSTANT /4/, while V.G. KAC /3/ has settled the classification of simple graded Lie algebras. Meanwhile a rather extensive amount of papers on graded Lie algebras was published in physical as well as in mathematical journals. In these notes we try to put graded Lie theory in the framework of duality theory of graded Hopf algebras. This seems to be of some interest, at least from an algebraic viewpoint. We are concerned with graded versions of some problems centering around the TANNAKA duality theorem and its analogues for Lie algebras indicated by G.HOCHSCHILD /1/ and /2/. 1. Let K denote a fixed base field of characteristic shall sometimes assume to be algebraically closed.

0 , which we

A linear space V is called g r a d e ^ if there are given distinguished subspaces and Vy, such that V = V^ © V y . The elements of the subspaces are called homogeneous^ and even or odd A according as they belong to V^ or Vy respectively. Let V and IV be graded linear spaces. The tensor product V