let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f being bounded Function of X, the carrier of Y holds (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f)
let Y be ComplexNormSpace; :: thesis: for f being bounded Function of X, the carrier of Y holds (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f)
let f be bounded Function of X, the carrier of Y; :: thesis: (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f)
reconsider f9 = f as set ;
f in ComplexBoundedFunctions (X,Y) by Def5;
hence (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms (modetrans (f9,X,Y))) by Def9
.= upper_bound (PreNorms f) by Th14 ;
:: thesis: verum