let X be non empty set ; :: thesis: for Y being ComplexNormSpace ex NORM being Function of (ComplexBoundedFunctions X,Y),REAL st
for f being set st f in ComplexBoundedFunctions X,Y holds
NORM . f = sup (PreNorms (modetrans f,X,Y))
let Y be ComplexNormSpace; :: thesis: ex NORM being Function of (ComplexBoundedFunctions X,Y),REAL st
for f being set st f in ComplexBoundedFunctions X,Y holds
NORM . f = sup (PreNorms (modetrans f,X,Y))
deffunc H₁( set ) -> Element of REAL = upper_bound (PreNorms (modetrans $1,X,Y));
A1: for z being set st z in ComplexBoundedFunctions X,Y holds
H₁(z) in REAL ;
ex f being Function of (ComplexBoundedFunctions X,Y),REAL st
for x being set st x in ComplexBoundedFunctions X,Y holds
f . x = H₁(x) from FUNCT_2:sch 2(A1);
hence ex NORM being Function of (ComplexBoundedFunctions X,Y),REAL st
for f being set st f in ComplexBoundedFunctions X,Y holds
NORM . f = sup (PreNorms (modetrans f,X,Y)) ; :: thesis: verum