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Additional info for A theory of cross-spaces
A in the from T^ Since clearly additive. 1. Since, Fd 8< V (S -. 2 furnish f cSc) = into *^x Conversely, every operator can be obtained in such a manner. / Afc)c The corre- K-8 Cy) is 50 CROSS-SPACKS OF OPERATORS III. ) llAll,,. This concludes the proof. 4. be interpreted as the Banach space of oO-norm of finite (with be approximated in that A Proof. ^ Let ot be a given crossnorm )) A 11^ norm all "X norm Then, A operators representing the . ^x from A ), Tr^ into T^ which may- by operators of finite rank.
For every pair Proof. 2. In into ( rnay be interpreted as the Banach <^,-norm, where of finite 7^> generates an operator ) A oL-norm and conversely. This correspondence into ")^*of finite II A // other words: ^ ^^7^ ( JT of TJ )** from is such that: (1) V <""* A (2)
1. a space of operators. a given o(/-norm. of finite + ; and <8d^ra, a s T^! " jp ^ A = If9 crossnorm OL^*X, the operators of finite The normed linear space rank^where || A || of operators represents the norm, may A from be characterized as Proof. A F. 1 to Af =2! o G- corresponds <) an operator F;(f)G a o - . ' . A in the from T^ Since clearly additive. 1. Since, Fd 8< V (S -. 2 furnish f cSc) = into *^x Conversely, every operator can be obtained in such a manner. / Afc)c The corre- K-8 Cy) is 50 CROSS-SPACKS OF OPERATORS III.