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13. is a ^ uniform crossnorm, that is, ^ satisfies con- ditions I--VI. Proof. I: Let2L* ( with F II l\ By Lemma = we 1 IJi* . 2 the F For an denote a fixed expression. 5 gives lence. II Zl7*F(fJg Now suppose F pose that = II is ^ ^(Zr= F|l g 1 . gj is a and such that Similarly we can find FC^with equivalent to Z^^tfjg^ > consequence 2^, h c k^ & <- Z^ f c * (2L^i f i . gj F)| By Lemma Naturally 8c = 1 . 5 for ^ . ) t Consequently, immediate and IV we can , is not . C "^*"for Let 2T7^, f c Ill: Clearly, || that JSE-jL, f^ there exists an l| - %> >Q 58 CROSSNORMS II.

2^*^ be fixed. 2 we have, g This clearly implies G (| G and <. )F(f)G(g)| II of expressions Therefore, the set of elements for in For any crossnorm Let Proof. dense is Then the set . 2 ou7i^V **c*i which these expressions stand LEMMA ,i= tt *i * denote two sequences of , elements dense in '^ and O^* respectively 2, 29 G ^^^C then also finite for every expression in also a norm. that the crossnorm 06^ is uniform, that satisfies the following condition: VI. ) for every pair of operators S <.

For our further discussion we , that is, a shall point out the precise conditions on the spaces the local character of b ) , a , shall need the following simple proposition: , a^^ denote positive numbers. n real numbers and Then, max b + Proof. Lemma The proof can be carried out easily by induction, verifying our first for n = 2 . *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.

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