Read e-book online Classical Banach Spaces I: Sequence Spaces PDF

By Joram Lindenstrauss

ISBN-10: 3540377328

ISBN-13: 9783540377320

ISBN-10: 3540606289

ISBN-13: 9783540606284

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Extra resources for Classical Banach Spaces I: Sequence Spaces

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There are also examples of spaces which fail to have an unconditional basis but do embed into a space having such a basis. The first and perhaps simplest example of this kind (cf. [82]) is the subspace D of ft, spanned by xn=en-(e2n+e2n+l)/2, n=l, 2, ... , where {e n}:'=l are the unit vectors in 11' The sequence {Xn }:'=1 forms a monotone basis of D. 12 since D is weakly sequentially complete without being isomorphic to a conjugate space (this latter fact will be proved in Vol. IV). More complicated but also more interesting examples are obtained by using Enflo's solution to the basis problem (and its modification in [20] and [40]); there exist subspaces of Co and 11" 2

But fails to have a basis? P. is the one given by the following result of [118] and [61]. 13. P. isomorphic to a complemented subspace of a space with a basis. if and only if X is Proof [118]. The "if" part is trivial and so we have merely to prove the "only if" part. We start by making two observations. 1. P. Then there exists a sequence QQ of finite rank operators Itt s,1I ~ {Sn}~=l on X so that X= A. for every n. Indeed. let {yj}~ 1 2: Snx. for every x EX. and n=l be a dense sequence in X. 2.....

Max an u, for every a, and take as Ll the family of all o={an };'=1 which satisfy 1 above and for which U1 = 1 and an +1 =ifi(a1 U a2 U ...

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Classical Banach Spaces I: Sequence Spaces by Joram Lindenstrauss


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