let X, Y be ComplexLinearSpace; :: thesis: the carrier of X --> (0. Y) is LinearOperator of X,Y
set f = the carrier of X --> (0. Y);
reconsider f = the carrier of X --> (0. Y) as Function of X,Y ;
A1: f is homogeneous

proof

let x be VECTOR of X; :: according to CLOPBAN1:def 3 :: thesis: for r being Complex holds f . (r * x) = r * (f . x)
let c be Complex; :: thesis: f . (c * x) = c * (f . x)
thus f . (c * x) = 0. Y by FUNCOP_1:7
.= c * (0. Y) by CLVECT_1:1
.= c * (f . x) by FUNCOP_1:7 ; :: thesis: verum

end;

f is additive

proof

let x, y be VECTOR of X; :: according to VECTSP_1:def 19 :: thesis: f . (x + y) = (f . x) + (f . y)
thus f . (x + y) = 0. Y by FUNCOP_1:7
.= (0. Y) + (0. Y) by RLVECT_1:4
.= (f . x) + (0. Y) by FUNCOP_1:7
.= (f . x) + (f . y) by FUNCOP_1:7 ; :: thesis: verum

end;

hence the carrier of X --> (0. Y) is LinearOperator of X,Y by A1; :: thesis: verum