let X, Y be ComplexNormSpace; :: thesis: for f, h being Point of (C_NormSpace_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 Point of (C_NormSpace_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) )

reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;

reconsider h1 = h as 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 Point of (C_NormSpace_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) )

reconsider f1 = f as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;

reconsider h1 = h as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;

A1: now :: thesis: ( h1 = c * f1 implies h = c * f )

assume
h1 = c * f1
; :: thesis: h = c * f

hence h = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f1] by CLVECT_1:def 1

.= c * f by CLVECT_1:def 1 ;

:: thesis: verum

end;hence h = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f1] by CLVECT_1:def 1

.= c * f by CLVECT_1:def 1 ;

:: thesis: verum

now :: thesis: ( h = c * f implies h1 = c * f1 )

hence
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
by A1, Th24; :: thesis: verumassume
h = c * f
; :: thesis: h1 = c * f1

hence h1 = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * f1 by CLVECT_1:def 1 ;

:: thesis: verum

end;hence h1 = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * f1 by CLVECT_1:def 1 ;

:: thesis: verum