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Theorem 2J. <8f~"^ is a proper subspace of completely continuous operators from T^ into 1^, would answer negatively the "basis problem" i, p. Ill] . Suppose A is a completely continuous operator which cannot be approximated in bound by operators of finite rank. separable Banach space The range '^ Q, of p. 96, A may be considered contained in a Theorem l] . It is well known that a completely continuous operator on a Banach space whose values lie in a 52 CROSS-SPACES OF OPERATORS III. 7. Let proper sub space of Proof.

Then of > Let 2T,ft shall prove our bers (Definition (a) on ^^^^ is an extension 7 on Proof. We ")l in the sup > gc Lemma, be an expression in by showing the equality 7^0771^ C. ^T^ of the following . *, G e 7^5 6. )| Hahn-Banach*s extension theorem an additive and bounded functional F [1, p. I|F 55] = JJ G Ji which states that on a linear manifold "W^ in a Banach space 1? can be extended (quite often in a non-unique manner) to an additive and bounded functional F on TC> , without affecting the value of the bound, .

2 . 4. 7. CHAPTER II CROSSNORMS The normed linear spaces 1. ^fiG^c'c^*. Henceforth, we assume that T>and \f while "^ and ^^denote any two Banach spaces ^f and 1^ stand for their conjugate spaces, that is, the spaces of all additive and bounded functionals on T^and*")^ respectively. Clearly, an expression ^> an operator A of finite ^ g^ in f. 8, also every operator A of finite rank. of finite rank many such expressions. To mined by where assumed ajF) = may sum space this if ^ for F In the case when *^j from T^Jinto 72 is e , ......

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A theory of cross-spaces by Robert Schatten

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