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Additional resources for Bundles of Topological Vector Spaces and Their Duality
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E. implies the 9j(~ - p) 6. 3 Corollary. a compact base C(X)-submodule Let space of p : E ~ X be X and F(p). this theorem. 9) we h a v e spaces maps In and Now suppose map 6A which F is dense completely find an answer: in r(p). metrizable : F(p) + FA(p). is a q u o t i e n t map. spaces over dense D of our S t o n e - W e i e r s t r a B F(p) regular base the e v a l u a t i o n maps. t h a t A c X is any subset. 4 Theorem. of seminorms complete. Let and If A extended ÷ FA(p) Proof. 6) EA(O') c Cb(X) in image FA(p) : of EA is s t a l k - for a c e r t a i n function of using in the p r o o f s .
0) Ul,,n I B belongs to such that = p(~'). 5 (B - let provided o U~, n. B' we c W, as a neighborhood is can the find only n UI, n that : X ÷ easy to c Vl,n}, Since if W' it a neighborhood see base that I £ A and mapping add a neighborhood ~ and is n are c p(W' continof enough. a neighborhood Vl, c Ul,,n n P-I(p(W' I Moreover, Now E be of e, we B £ Ul iinl- n U l n ). , there Ul, n is This n EvE) a selection. e. B' c W, Let element 61-n-thin implies n is n Ul,n)). e. ) o is s-n-continuous, that The mapping (ii) For every o E > O : X + and ~.
6(viii)). continuous• E J such neighbor- is an e l e m e n t is a t o p o l o g i c a l ÷ E are j" x O, c 0 E itself an o p e n find = fi(°1(x O) .... ,oT(i)),a',j') apply the we may neighborhood 0 = T(U,o,e,j). e. is inequalities Oj. (<~T (i) - a ' T (i) ) < 6 9j, (a I - a~ ) ..... imply 9j, (fi(al ..... qT(i)) We show that fi T(i) : V S * ~ maps T(U',~I,6,j") of x x .... x T ( U ' , o (el .... 'e~(i) ) i n t o Indeed, let E with Whence • .. = p ( B T ( i ) ) 8k = o~(x) we may and conclude the open < £' neighborhood (i),6,j") n T(i) V [ 0.