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 = sup (PreNorms f)
let Y be ComplexNormSpace; :: thesis: for f being bounded Function of X,the carrier of Y holds (ComplexBoundedFunctionsNorm X,Y) . f = sup (PreNorms f)
let f be bounded Function of X,the carrier of Y; :: thesis: (ComplexBoundedFunctionsNorm X,Y) . f = sup (PreNorms f)
reconsider f' = f as set ;
f in ComplexBoundedFunctions X,Y by Def5;
hence (ComplexBoundedFunctionsNorm X,Y) . f = sup (PreNorms (modetrans f',X,Y)) by Def9
.= sup (PreNorms f) by Th17 ;
:: thesis: verum