By Robert Schatten
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Extra resources for A theory of cross-spaces
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. 4 give, The ^ last inequality holds for any V: - F(f) Let llf :, and II f gc c llF C^ = II f 1 g We . p. 55] ["l, This proves . choose an By Lemma . III. 5 furnishes VI: Let T and S ^ ( f 8 = ) H fH K g // denote two operators on T& and 7i respectively. fll we are running through extreme = ) k ' a implies T^ 2-^, ^T Sh> Tk ') This concludes the proof.
T gj c and onlv if = DEFINITION F. 1. is * f-S g^ in (^g I . O ^ gc ^o + Whenever IV; . a <*/( . cU2f) 21^ F. ^ For a . ) O as satisfying the inequality: "^^. 2^. expressionZ^
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. 3. 3. f the following "dual" of the last statement: . , Then, Let^^RflTG, JSE FC G^ T^i be an expression in J^fe represents an expression in 6, S^ KJ im P lies /* while Under their "inner THE ALGEBRA OF EXPRESSIONS I. product" in symbol (2:^j 8 F. ^ ( ^ JZT< * A * gc. 4. The inner product is invariant under equivalence. We hereby, that K.
A theory of cross-spaces by Robert Schatten