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 = upper_bound (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 = upper_bound (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 = upper_bound (PreNorms (modetrans (f,X,Y))) ; :: thesis: verum