Lp-Structure in Real Banach Spaces - download pdf or read online

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By Ehrhard Behrends, Rainer Danckwerts, Richard Evans, Silke Göbel, Peter Greim, Konrad Meyfarth, Winfried Müller (auth.)

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I~. fY is thus in LP(m x) ( ~ S ( ~ ;x) ) and llfyI~ Y7 Pdm = ~ l~II- We define a m a p p i n g F from X into S ( ~ ;x) by means of y ~ fY. 51 Since x' is l i n e a r If g E C ( O ) and has n o r m then~oh other hand x'(hgx) I, F is also l i n e a r fgx dm x = x'(hgx) = and llFli-

It f o l l o w s On the that fgx = g and thus that Fy = y for all y in S ( ~ ;x) of the form f = gx for some g in C ( O ). H o w e v e r elements of this form are dense by continuity, Fy = y for all y in S ( ~ ;x). ~hfgYdmx=X'(hgy) with (lin ~ )-, a maximal for all h E C ( ~ ) and on the other for all h E C( ~ ), g f Y = f g Y and F c o m m u t e s w i t h the p r o j e c t i o n s of their L P - s t r u e t u r e , algebra also g e b r a implies for g E C ( ~ ), y E X on the in ~ . is the key to our c h a r a c t e r i z a t i o n Boolean t h e o r e m must ;x).

U), to so that (i) - is a p a r t i c u l a r l y conditions. disconnected on K. Then, ever3~#qhere space compact topological every m - m e a s u r a b l e to some c o n t i n u o u s func- numerical function. Proof: Consider class containing the map C(K) f). ~ L°°(m) d e f i n e d by f ~ (the e q u i v a l e n c e Then a) This map is l i n e a r b) If If(k) l > r for some k then since open set 0 w i t h k E K on w h i c h If(1)l open set has p o s i t i v e m - m e a s u r e , norm in C(K) L~(m) c) If ) and the almost ) are identical, n ~ riXBi i=I set w h i c h f is c o n t i n u o u s the absolute everywhere i.

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