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4). A pair ( E , p ) consisting of an algebra E and a submultiplicative semi-norm p on it is called a semi-normed algebra , while E is thus considered to be topologized as before. Thus one concludes by ( 3 . 3 ) that every semi-normed algebra ( E , p ) i s a metrizable l o c a l l y m-convex algebra. Moreover, a topological algebra E whose topology can be defined by a submultiplicative semi-norm is called a semi-normable algebra (cf. 1). Now on the basis of the situation one has in the case of topological vector spaces, one concludes that a semi-normed (resp.

EXAMPLES OF TOPOLOGICAL ALGEBRAS 9 An i n - b a r r e l l e d t o p o l o g i c a l a l g e b r a which i s , i n p a r t i c u l a r , c a l l y convex ( r e s p . , l o c a l l y rn-convex) , lo- w i l l be c a l l e d an rn - bcrrelled l o c a l l y convex ( r e s p . l o c a l l y m-eonvex ) algebra. I n t h i s r e s p e c t , w e remark t h a t t h e l a s t t h r e e c l a s s e s of t o p o l o g i c a l a l g e b r a s are i n d e e d c o n w i t h an o b v i o u s " i n c l u s i o n r e l a t i o n " ; t h i s i s , i n e f f e c t a ge- nected one, a s it w i l l be shown by t h e Examples of t h e n e x t s e c t i o n .

Moreover, t h e r e s u l t i n g c l a s s e s of t o p o l o g i cal a l g e b r a s a r e n o t n e c e s s a r i l y t h e same, a s w e s h a l l see below ( c f . Example 2 . 4 ) . 2. Examples o f topological algebras W e g i v e below some f i r s t examples of t o p o l o g i c a l a l g e b r a s which s e r v e t o c l a r i f y t h e nocions a p p l i e d h i t h e r t o and, i n p a r t i c u l a r , t h e way one a p p l i e s a number of t h e p r e c e d i n g r e s u l t s ( f o r example, Theorem 1 .

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