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

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Space =: sup E T may let F be ~ the mapping < e. T h i s u ]2~ - PT with usual the plication with -I), PT If F = ~ 2 with the Euclidean : ET + \$I norm : ET ÷ \$I metric such S I is z = I. T h u s , we o(~) II can < £. e. Moebius if T = is < e and implies T = -I is t h e requires e. 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).

AT(i)) then of t y p e = \$ ( f i ( a l ..... a T ( i ) ) ) - a E is c a l l e d - b. e N for a l l 3 3 ~-spaces ~-morphism is an a reverse of (topological) of E. Then E/N the a d d i t i o n a l between n-ideal. 2): ~-space is also operations a of type T and (topological) (fi)i£I on E / N 81 are defined by fi(al + N .... ,aT(i) + N) := fi(al ..... aT(i)) + N. Moreover, the canonical quotient map ~ : E + E / N is a (continuous and open) ~-morphism. 4 isomorphic. D Let ~ : E ÷ F be a Proposition.

Are due has The results to M a c h a d o , generalized [We 65], which Nachbin will be and P r o l l a ([Ho 75]). : E ÷ X be a b u n d l e . A family if e v e r y point {i : fi(y) for some y ~ O is c a l l e d belongs every x c a neighbor- is finite. oi been [GI 63], locally finite, F c F(p) family for b u n d l e s locally (oi)i• I c F the that this selection is bounded. A subspace the set F c F(p) £x(F) is d e n s e the e v a l u a t i o n It is o b v i o u s is c a l l e d in the stalkwise dense, stalk Ex, where if for e a c h ex : F(p) x • X ÷ E x denotes map.