Lp-Structure in Real Banach Spaces by Ehrhard Behrends, Rainer Danckwerts, Richard Evans, Silke

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

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Then ~ rix D is c o n t i n u o u s i=I i n e q u i v a l e n c e class c o n t a i n i n g i=I~riXBi functions. > r. Since there are dense includes = 0 for all all step in L°°(m) the map- 82 For every n u m e r i c a l by f+(k) function := f ( k ) / ( 1 + i f ( k ) I ) tion g from K into g(k)/(1-ig(k)l) Suppose bounded m-measurable let g- be the f u n c t i o n -+1/0 f is a n u m e r i c a l g-(k) := := + m-measurable function nuous function with norm defined w i t h -+~/(I+°°) :=-+I and for every func- [-I,+I] whereby f on K let f+ be the f u n c t i o n function on K.

L is a n o r m r e s o l u t i o n that to C( P ). In the fore- In section the concept Z is an integral chapter exist perfect m e a s u r e s in cP( 0 ;m). of a f a m i l y seen in the p r e v i o u s isomorphic by virtue norm resolution of the X k with as of ~ ). for X taking v a l u e s in c P ( ~ ;m). T h e n as n o t e d at the end of section C, for every k a) [x + y](k) S [xT(k) b) [rx](k) Consider = Irl[x](k) + [yT(k) for all x,y in X for all r in L, x in X. for some k the set Yk := (x I x E X, [x](k) by a) and b), Yk is closed u n d e r addition ~ ~ } c X.

J~ hg dm x. 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 ~ .

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