By J. Bourgain, P. G. Casazza, J. Lindenstrauss
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Extra info for Banach Spaces With Unique Unconditional Basis, Up to Permutation
This diametrically opposed behavior of the natural bases of X and Y shows that X fails to have a unique unconditional basis, up to permutation. We consider now the space Z = ( £ 1 ® £ 1 ® ' - - 0 £ 1 ® - • • )2- It is easily seen that Z is isomorphic to X®Z and thus also to Y@Z. If Z would have unique unconditional basis, up to permutation, then the natural basis of Y © Z would be equivalent to a permutation of the unit vector basis of Z. In particular, the natural basis of Y would be equivalent to that of a subspace of Z of the form ( £ ®0>\j)z with 1 ^ kk < <», for all j .
E. 1, for all m and ;. ozi the empty set corresponds the the only function if the maximal if £ = 0). Clearly, the maximality of \ik\$~v we > c» where I * 3-(g+1>, for some 1 ^ m ^ 2*\. Since v is a probability measure and there are n - £ possible indices i it follows that there is a j 0 which belongs to at least e(n —£) distinct s e ts of the form A^. Since m takes 2^ values it follows that there is an m 0 and a set 6X c [l , . . *,) Since d s£ 2 1 2 A 4 (logn) 2 / 3 it follows that, for n > n 0 (A,p), and, consequently, [(n-£)£-2-*-1]'>3g+1 Obviously, also for large n , <$>logn.
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 ^. 6. 1 yields immediately the following infinite dimensional result.