By Gerhard Gierz (auth.)
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Additional info for Bundles of Topological Vector Spaces and Their Duality
9 If f l , . . , f n : X + [O,I] c]R are continuous functions such n that ~ fi(x) = I a n d if a l , . . , a n : X + ~ a r e ~ - n - c o n t i n u o u s i=I n selections, then ~ f . a . is a f - n - c o n t i n u o u s selection, too. 10 Let uous lim e > O and selections 0n(a m every - n' s (Firstly, o) let and (Gm)me ~ let = O. e. s-n'-continuous of e-n-contin- that selection for n. there 9n(0 m Fix : X + c is a positive - 0) < s/2 Vn(Om(X) - o(x)) let U be neighborhood an of open < 6/2 integer for neighborhood Om(Xo) such all of that m x that £ X.
1. Definition. a type. Let X be a topological If E is a t o p o l o g i c a l t h e n E is c a l l e d if the a x i o m vector We x c X, shall the (FM6) holds, provided L e t us which are c a l l e d o f all t h e n E is c a l l e d that the base start with is d u e the present situation: Let T : I + ~ (FMI) x (~j)j~j. an ~-function the v e c t o r stalks of t h e f u n c t i o n m o d u l e . module of a bundle space of is in f a c t ~-spaces - (FM5), Moreover, module of spaces D (isomorphic of the be same to) type, is c o m p a c t .
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 ~. ¢ ~, W' large of the n is ,n' E add(W'xOl, (i) If ~, is P(B) e. > n such c Of, n. 6 = and n' Then = form c W. ',n' have B - • ~} - Ul P(B) of n EvE) > I and that c U ll,nl. n e ~} Then base - We n < e I and neighborhood by l' : I £ A and : ~n(B) add(W'×Ol, open : I ~ A and VI, n = p(Ul,n).