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 Th17 ;
:: thesis: verum