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By I. M. Singer

ISBN-10: 0387048332

ISBN-13: 9780387048338

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For every pair of integers r > 1 and 1 < j < 771, we let ^rnr+j ke ^ translate of Uj so that, for h^ < h2, the support of uki precedes that of ukz. the integers, there Then, for every permutation TX of clearly 1 < j j < j2 < m = > r^ < Tjz exist and \rj\JL\ such wfaw^+Ji) < n(mrjB+j2). that Conse- quently, we get from (*) that m m TO >=i J=i i=i for every choice of scalars \a^\^Lv We can now follow Zippin's proof [37] verbatum and obtain the desired result. 1. Let I be a prime space with a normalized unconditional basis \xn\£=l which is unique, up to permutations.

A simple iteration argument shows that if \vi\t=\ are mutually disjoint blocks on fanJ^Li so that ||V{|| ^a, all 1 <£ i -s jf^f /i = 1,2, • • , and some a > 0, then for ii £^11 >o>*/«. i=l Finally, if {^i4i=i are mutually j h disjoint blocks on \un\™-x with h <>t < j +\ for some integer /i, and so tha t \\vi || > a, for 1 ^ i <> t, then II Et/tH > ^f 1 II £ v i l l > Kilajh'<* i =l i=l = K^rUqa>J{h+1)/q > Kf1]-1'***1"* Step HI. We are prepared now to prove that £/ has a lower p estimate for disjoint elements, for q

Assume that e = c/ (4K\\P\\). 1^(^)1 > c, for i = 1 , . . , s, Recall that, for a maximal subset \UJ\2=1 and put of vectors in u the unit ball #^(0,1) of X satisfying 11^— j z \\ > £» whenever j ^ ^ j 2 , we have p < (4/ e) dim (this follows \Bx(Uj,e/2)\f=1 trivially from x = (—^-li-^iL)dim x, volume considerations: the balls are mutually disjoint and all contained in Bx(0,2)). It follows from the form of s that we may assume that, for some j 0 , we have lla^-^Uj-JI < c, f o r i = 1,2 (+) m m, where >4K2c~2\\P\\2.

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Bases in Banach Spaces I by I. M. Singer

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