Classical Banach Spaces I: Sequence Spaces by Joram Lindenstrauss

By Joram Lindenstrauss

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Extra resources for Classical Banach Spaces I: Sequence Spaces

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Xjll~ 1, for every n, but "~l a"x" does not converge. It follows that there is an e> 0 and a sequence of integers Pl

1= 1 Conversely, assume that rp is a linear functional onL(X, Y) so that Irp(T)1 ~ CIITIIK' for some constant C and some compact set Kc X. 2 we may assume without loss of generality that K=conv {Xn}~=1' where IIxnll ~ O. Let S: L(X, Y) ~ (YE8 YEB ''')0 be defined by S(T) = (Txl> TX2'''')' Since Icp(T)I~CIIS(T)11 itfollows that there is a linear functional tP defined on the closure of SL(X, Y) so that rp(T)=if1(S(T». By the Hahn-Banach theorem we may extend tP to a continuous linear functional on (Y E8 Y E8 ..

For example, if Y has a Schauder basis {Yn}:'=1 then, for every compact TEL(X, Y), IIT-PnTIi ~ 0, where the {Pn}:'=1 are the projections associated to the basis {Yn}:'= l ' The question whether the converse assertion is true for arbitrary Banach spaces X and Y (which was called for obvious reasons the approximation problem) was open for a long time. This 1. Schauder Bases 30 problem was solved (in the negative) by P. Enflo [37]. The observation above shows that this solution provides also a negative solution to the basis problem.

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