By J. Bourgain, P. G. Casazza, J. Lindenstrauss

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Extra resources for Banach Spaces With Unique Unconditional Basis, Up to Permutation

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The operator defined by, 46 UNIQUE UNCONDITIONAL BASIS 47 n For each element of the form (p = 2 ^ ( 0 ^ in Zr 2 ([0,l],£P), put i=l i=l It is easily seen that P n is a projection from £ 2 ([0,l],£P) onto A^. The fact that | | P n | | ^ 2 1 / 2 can be proved in the following way, by using well-known properties of conditional expectations and the independence of t h e partitions jS'iJiLi1 n n * ll^nHI 2 = / ( E I ^ * i l ) 2 d ' = S / I ^ i l 2 d * + 0 1=1 n i=l 0 1 2 S/I^il i=l 0 n « + 1 i#J 1 i=l0 i#j x Zf\Ei

By duality, we get that || tea* Xbiu*\\<4C1M\\P\\2max\bi\, lGa * again for any system of scalars {b i { iea ^,l <>h < r . 2 is completed provided we show that r is bounded by a constant depending only on KltK2 and | | P | | . Put yZ = £ < and notice that ||y^ || ^ 4 6 ^ | | P | | 2 , for all 1 < /i < r . 6, II 2 < l l = II EVfcll ^ ( E n=l /i=l /i=l ll^ll p ') 1/p '^4C 1 A' 4 il/||F||2r^'. On the other hand, if i e crr then, by (b), we have (max|<|)(K|)^l/(2||P||), for every 1 < /i < r . )* >^ : r(2KI 2 ) 1 / 2 (KI)* v ^ : ill(2K/IKIII 8 ) 1 / 2 ll* n=l n=l > ^ Z I | | ( 2 l < l 2 ) 1 / 2 l l * C i ^ = r i l g<||<4C 1 2 ^ 4 i/||P|| 2 ri/^i/P', n=i n=i which yields that, r ^(leCf/^lfllPlI 2 )*'* 2 ^.

C,l 1/ ( 2 | | P | | ) , for all 1 :< ft < £, then we assign i to a new subset a# + 1 . Otherwise, there exist integers h so that l < / i < £ and ( m a x | ^ | ) ( | u t |) < 1 / ( 2 | | F | | ) . Let /i 0 be the least integer satisfying this condition.

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