By Joram Lindenstrauss
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Extra resources for Classical Banach Spaces I: Sequence Spaces
Example text
Biorthogonal Systems The existence of separable Banach spaces which fail to have a basis motivates the attempts to try to use some weaker forms of coordinate systems. One approach, which has been studied for a long time and for which strong existence theorems are now available, is that of using biorthogonal systems. U. Let X be a Banach space. A pair of sequences {Xn};:"=1 in X and in X* is called a biorthogonal system if x;(xn)=S:i'. A sequence {Xn};:"=l in X is called a minimal system if there exists a sequence {X~};:"=1 in X such that ({xn};:'=l> {X~};:"=l) is a biorthogonal system.
First we notice that L(l~, X**) =L(lr, X)**. This follows from the fact that the correspondence T_ {Yt=Tei}f=l (where {eaf=l are the unit vectors of n times m is an isometry from L(n, X) onto (X EEl ••• EEl X) co. Let I: D - X** be the identity mapping and let e> 0. There exist an n and vectors {Uj}7 =1 in D of norm ~ I + e so that conv {uj}f =1 ::::> B D(O, I). Hence, there is an operator' V:lr -D, for which 11V11~I+e and VBI~(O, 1)::::>BD(O, 1). Since IV EL(lr, X)** there is a net {S,,}cL(I~, X) with IIS"II~ IIIVII ~ 1+e for all a and {Sa} converges to IV in the w* topology of L(l~, X)**.
This was shown by T. Figiel and W. B. Johnson [41]. P. P. if, for every finite dimensional subspace B of X and every 13 > 0, there is a finite rank operator T on X so that IITx-xll:(e+13)llxll for XEB and IITII:(A+13. P. P. P. P, for some N = A/CA, e). 1S. P. P. ° Proof Let 13 > be such that e + 13 < 1 and let Bc X with dim B <00. inductively a sequence {Tn}:~ 1 of finite rank operators of norm :( A+ 13 on X so that IITI x-xii :(e+ 13)llxll, for x E B, and IITn+lx-xll:(e+13)llxll, forxEspan{BU Q TiX}' For n"?