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By Dale E. Alspach, William B. Johnson (auth.), Ron C. Blei, Stuart J. Sidney (eds.)

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Read or Download Banach Spaces, Harmonic Analysis, and Probability Theory: Proceedings of the Special Year in Analysis, Held at the University of Connecticut 1980–1981 PDF

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Extra info for Banach Spaces, Harmonic Analysis, and Probability Theory: Proceedings of the Special Year in Analysis, Held at the University of Connecticut 1980–1981

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Pisier's on tensor products. seems to be a very explains It also of Lq(~,P) and the simple seems to be quite theorem (q > 2, L2-norm C(X)-space Grothendieck's of this theorem theorem. illuminating are equivalent, measure) then every it p r o o f of that in the sense In one way or a n o t h e r a probability and we be- If so, are based on the fact that if P is Our n e w p r o o f of P i s i e r ' s (not to say simple-minded) why other proofs work. Grothendieck's contains also a p r o o f of G r o t h e n d i e c k ' s that it is new also as a proof of P i s i e r ' s Since a theorem lieve theorem.

The measure assume that so we shall prove that if p = fdl , f,g ~ L2(X,dl) v = gdl I = v (Pl+Iv), we may then sup IIf + e i e g l d l 0<8<2~ ~ (lifIl~ + ~ 2 1/2 IlgllI) Let us start from the left. sup fl f + e i e g l d l 0 i---~ Ilf(x) + e i e g ( x ) Idl(x)de -- 2~ o = I 2~2/~ I f ( x ) + e l S g ( x ) IdOdl(x) X o (as is easily seen) 2~ : XI ~ °/ lJf(x) l + e i8 Ig(x)lldedl (x) = I 1 ~ ilf(x) I + eielg(x ) II dedl(x) X~ o 1 ~ >_ I xI ~ o~ (If(x) l + +~. i llflll and this proves Sublemma 8 2.

Corollary. We may write Nj ~(z) where degree mn K. is a m o n o m i a l = ~ mn(Z) ~n(Z) n=l of d e g r e e K-k and ~n is h o m o g e n o u s of Further 2 [I[~nll2 ~ -1/211~ll ~j Splitting using up the inner product the e s t i m a t e on l[sll op according we have to this the decomposition required and result. I should l i k e to take this oppo~uni~y to thank the U~versity of Connecticut for i t s hospitality d~ing the special year in Harmonic Analysis. 32 References [i] [2] [3] J. V o n N e u m a n n , Eine eines Raumes, unit~ren S.

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