By Gerhard Gierz (auth.)

ISBN-10: 3540116109

ISBN-13: 9783540116103

ISBN-10: 3540394370

ISBN-13: 9783540394372

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**Extra info for Bundles of Topological Vector Spaces and Their Duality**

**Example text**

E = S I, (FM4) : z means IITII Ilo(o) II a bundle then axiom problematic ~ c [0,6[ I~l is t h e space I ~O) ' (O = E T. T is m u l t i - strip. then 56 PT : ET ÷ $I More is h o m O t o p y generally, equivalent if F = ~ n with to K l e i n ' s the E u k l e d i a n and operator with dimensional analogon of the M o e b i u s s t r i p and K l e i n ' s bundle PT tions). 15 of some fixed is in fact the b u n d l e s Sequence family of seminorms seminorms given n c ~. finer than the c o f i n i t e Co(E) is a f u n c t i o n set {n E ~ e > O, : ~ j ( u n) < O.

Because in a bundle. already is locally countable Definition. in like to use bundles as spaces unsatisfactory given base is the p o s t u l a t e we w o u l d spaces paracompact locally axioms. para- and p a r a c o m p a c t neighborhood. It can be shown regular. that every On the other paracompact space paracompact. 2 Theorem. directed family topologies such locally locally has a n e i g h b o r h o o d result paracompact is stated is c o m p l e t e l y space, space paracompact space every is locally every p o i n t sets.

6 T(V,Ol,6,j)×T(V,o2,6, all p(a) j) that for n EvE all is c T(V,~I,6,j)×T(V,a2,6,J) _< ~ j ( o L ' - Crl(p(ot'))) + Vj ((O I + 02 < 6 + = 6 + E- 2"6 y e V. a neighborhood n EvE we + x)j(8' of obtain - 02(p(B'))) - O)(p(a'))) 48 This Now shows let r o . ~ O. the (ro,eo)£ In element this o' vj(r ° continuity ~xE case ,0! (p(~o)) and we £ E such of let T(U,o,E,j) choose that - add. 6 = be open E - ~j(o(p(eo)) r o . ~ j (o' (p(~o)) o(p(~o))) an - s O) ! _< ~ j ( r o . So). < 6/2. (e' + 9 (r'e 1 < E' o - ] O < e' (o')) and 0(p(~))) - if V o(y)) such o - c U of Pick we an obtain + a(p(~o))) < E - p(eo ) such 6/2 for all 6/2) < ~.

### Bundles of Topological Vector Spaces and Their Duality by Gerhard Gierz (auth.)

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