By Anastasios Mallios

This quantity is addressed to people who desire to practice the equipment and result of the idea of topological algebras to various disciplines, even supposing faced by means of specific or much less normal kinds. it can even be of curiosity to people who want, from a wholly theoretical standpoint, to work out how a ways you'll be able to transcend the classical framework of Banach algebras whereas nonetheless keeping sizeable effects. the necessity for such an extension of the normal concept of normed algebras has been obvious because the early days of the idea of topological algebras, such a lot significantly the in the neighborhood convex ones. it truly is worthy noticing that the former call for was once due not just to theoretical purposes, but in addition to strength concrete purposes of the hot self-discipline.

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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 .