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Of the greatest It is . 1. , p. 138, for of termed a is complementary manifold. 2. projection in a same time an ' not smaller than the value for that T*> ( O ^^ . 54 CROSS-SPACES OF OPERATORS III. 6. ** such that T/= Ttt, 7u is the conjugate space of Suppose further . ) denote a Banach space and *WT a closed linear "Y^ that, TTU,. C T^'W, crossnorm on ^^uLc without changing which (fg) = =|||il|| = ^r ffcHT, 8 eTTLo , and .

ZT^ vSf i* Proof. For a given expression JS'IT ^L^ 8, ^gJ the same (i)--(iii). = JE"J^, F(fjg c Since . for both sides of any of the This means F . f cl^i we form ^ o implies the transformation the values of ^** relationships k. 4 T remain expressed by Rules that a single application of these Rules to an expression T does not change the value of . Thus, a finite number of successive appli- cations of Rules (i)--(iii) to an expression does not change the value of for that expression. fl| g >% This concludes the proof.

K> CROSSNORMS II. 4. f be a functional of expressions on 1j satisfying the following conditions: assumes For equivalent expressions O (i). \&(*Z (ii). 6 Then, " *^* tf2, Proof. Since < ( *,<* f ~ . 13, V) the right side of our equality is clearly not greater then the left side. 2^^ (i) and be fixed. -)l max expressionZlT^ ff Now let C^^T_ gf f. 14, the '*^** value. t t sup same the is . 5 implies II The last inequality holds. for every expressionS^ifj the left side of the equality of our theorem is g^ .

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