let X be non empty set ; for Y being ComplexNormSpace
for f, h being VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
let Y be ComplexNormSpace; for f, h being VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y))
for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
let f, h be VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)); for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds
for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
let f9, h9 be bounded Function of X, the carrier of Y; ( f9 = f & h9 = h implies for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) ) )
assume A1:
( f9 = f & h9 = h )
; for c being Complex holds
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
let c be Complex; ( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
A2:
C_VectorSpace_of_BoundedFunctions (X,Y) is Subspace of ComplexVectSpace (X,Y)
by Th7, CSSPACE:11;
then reconsider f1 = f, h1 = h as VECTOR of (ComplexVectSpace (X,Y)) by CLVECT_1:29;
A3:
now ( h = c * f implies for x being Element of X holds h9 . x = c * (f9 . x) )end;
now ( ( for x being Element of X holds h9 . x = c * (f9 . x) ) implies h = c * f )end;
hence
( h = c * f iff for x being Element of X holds h9 . x = c * (f9 . x) )
by A3; verum