let X, Y be ComplexNormSpace; :: thesis: 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)); :: thesis: 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; :: thesis: ( 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;

then h1 = c * f1 by Th16;

hence h = c * f by A1, CLVECT_1:33; :: thesis: verum

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)); :: thesis: 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; :: thesis: ( 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 :: thesis: ( ( for x being VECTOR of X holds h . x = c * (f . x) ) implies h = c * f )

assume
for x being Element of X holds h . x = c * (f . x)
; :: thesis: h = c * fassume A2:
h = c * f
; :: thesis: for x being Element of X holds h . x = c * (f . x)

let x be Element of X; :: thesis: h . x = c * (f . x)

h1 = c * f1 by A1, A2, CLVECT_1:33;

hence h . x = c * (f . x) by Th16; :: thesis: verum

end;let x be Element of X; :: thesis: h . x = c * (f . x)

h1 = c * f1 by A1, A2, CLVECT_1:33;

hence h . x = c * (f . x) by Th16; :: thesis: verum

then h1 = c * f1 by Th16;

hence h = c * f by A1, CLVECT_1:33; :: thesis: verum