By Kenneth I. Joy

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Space is topological. to 4g In case X is a locally compact space, then initial topology generated by all sup-seminorms, Cc(X) carries the that is by seminorms of type sK c(x) > m f where K c X ) suplf(p)l ps is a compact subset of X. , Thus Cc(X) carries in this case the topology of compact convergence. To prepare for the study of locally compact spaces we first focus On~cE , the dual space ~ E dowed with the continuous of a t o p o l o g i c a l convergence s t r u c t u r e m-vector space E en- U of (see [ Sch I ]).

For any n e i g h b o r h o o d is a compact t o p o l o g i c a l subspace of ~cE; in fact it carries the topology of p o i n t w i s e convergence. Proof: Let ~ be an u l t r a f i l t e r on U~ with the topology of pointwise o Us to some f u n c t i o n a l to f U~ s' convergence, the polar is compact We proceed to show that r U~ 9 endowed converges in even converges in ~ c E. For any element T E | f. Since e s E and any p o s i t i v e real number ~ we find a with ~(T Hence for any k(e + k s T 9 U) c x {e}) c f(e) + [ r 2 2 we have k(e) + ~s 9 k(U) ~- f ( e ) + [- c , ~E ] ~ + -2 9 [-1,1].

X is called compact if F is A c o n v e r g e n c e space is said to be locally compact if it is H a u s d o r f f and every convergent filter contains a compact set. As in topology compactness for convergence characterized spaces can be in terms of "coVerings". A system $~ subsets of a c o n v e r g e n c e space ring system if each convergent filter on X X is called a cove- contains some element of ~ . e. W is the limit of a filter any limit of a filter ~ on X X A point p s X or a point of adherence of >p) finer than is adherent to ~.

### Bernstein polynomials by Kenneth I. Joy

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