# Download C* -Algebras Volume 1: Banach Spaces by C. Constantinescu PDF

By C. Constantinescu

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K---~cx~ T h u s # is a measure. Take A E 91 and c > 0. Take n E IN w i t h n >__p~. Since #n is a R a d o n m e a s u r e on T , there is a K E ~ , w i t h K C A a n d 30 1. Banach Spaces I#~(L)- #~(A)I < for every L C N , K C L C A . T h u s I~(L) - ~(A)l <_ I#(L)- ~(L)[ + I#~(L)- #~(A)I + I#~(A)- #(A)I < for every L E N, K C L C A . Hence # is a R a d o n m e a s u r e on T . Take c > 0 a n d n E ]IN with n > p~. T h e n I#- ~nl(T) _ 4~, so t h a t # - #n C Adb(T) and II~- ~11 ~ 4c. Hence # E 3rib(T) a n d lim #n = #.

C => a. Let (tn)nE~ be a sequence in A. ) < - n for every n E IN. ),E~ which converges in T . Let 20 1. Banach Spaces t'- lim sk~. n---+ (:XD m Then t E A and lim tkn = t . n---+ (x) m Hence A is compact. 12 t 0 ) Every relatively compact set of a metric space is precompact. The converse implication holds whenever the metric space is complete. m Take a relatively compact set A of the metric space T. 11 b =~ a). Hence A is precompact. Conversely, let T be a complete metric space and A a precompact set of T.

TcT b) If each Et (t 9 T) is complete, then Fp is also complete. c) If p r oo, then {x e Fp I {t 9 T I x ( t ) ~ 0} is finite} is a dense vector subspace of Fp. d) Fo is a closed subspace of Foo. 5. b) Let (Xn)nc~ be a Cauchy sequence in Fp. ) < e for m , n C IN with m_> r n ~ , n _> m~. Then, given t E T , ( x . ( t ) ) . ~ Cauchy sequence in Et . (t) n---~ (x) for every t C T. Then x ~ - x C Fp and is a 18 1. Banach Spaces q(xn - x) < c for every c > 0 and n > rn~. Hence x C Fp, (xn)ne~ converges to x, and Fp is complete.