let X, Y be ComplexNormSpace; for f, h being VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y))
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let f, h be VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)); for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let c be Complex; ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
A1:
C_VectorSpace_of_BoundedLinearOperators (X,Y) is Subspace of C_VectorSpace_of_LinearOperators (X,Y)
by Th21, CSSPACE:11;
then reconsider f1 = f as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by CLVECT_1:29;
reconsider h1 = h as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;
hereby ( ( for x being VECTOR of X holds h . x = c * (f . x) ) implies h = c * f )
end;
assume
for x being Element of X holds h . x = c * (f . x)
; h = c * f
then
h1 = c * f1
by Th16;
hence
h = c * f
by A1, CLVECT_1:33; verum