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Every compact subset norm f —►llfllj^ top olog y « on fam ily E; i f on 15(E) KCE E = já, we set 1S(E;K) simply by determines an algebra semi­ i ( E ) . pact-open is the topology r o f such algebra seminorms. algebra, by P roposition 2 , i l 3 . iSp defined by the d irected 15(E) i s thena to p o lo g ic a l -s REMARK 1 “ Assume ■¿(E) E to be compact. Then the topology defined on by the sin g le algebra norm ly the natural topology of remark that |f llj^ ^ l|f llj, if f l|f Hg is p r e cis e ­ ti(E) ; th is fo llo w s re a d ily from the K CE a l l to p o lo g ic a l considerations on and TS(E) f e 51^(E) .

Let and KCR^ a f e ) compact be given; there i s |p(x) - f(x)| < e if and Í^(R^^;|K) of is dense in the subset a. This fo llo w s from f ( a ) = 0. Let p e P(R^^;IK ) e>0 such that xeK. There is no lo s s of g en era lity in assuming aeKj then |p(a)| < e. L etting q(a) = 0 and |q(x) - f(x)| < 2e fo r q = p - p( a) , we have xeK, as wanted. We have deduced the above W eierstrass theorem from the W eierstrass-Stone theorem. Notice that Lemma 1 ,H 7 is a very s p e c ia l case o f the W eierstrass theorem on the re a l lin e .

If x ,y e E, x/^y, there i s ie l , and W = {teE |l f i ( t ) - f j^(y)l< 6/ 2 ), where Then V and i s a Hausdorff be orthonormal to j a . l . such that con d ition (3) o f above D e fin itio n . Let E fj^(x) f fj^ (y), by V = {teE ||fj^(t)-f^(x)l< 6/ 2 } 6 = |f^(y) - f^(x)| > 0. W are d is jo in t neighborhoods of x and y, resp ect­ iv e ly . DEFINITION 4 - Let E be a to p o lo g ic a l v e cto r space. ia^}. ‘ neighborhood any f i n i t e V J C I, _ c: e lel of is summable with sum -------- 0, there i s a f i n i t e , J3J^ 'w e have ^ aeE i f , fo r any such th at, f o r V.

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