linear algebra, inner product spaces

Let V be a complex inner product space and T a self-adjoint linear operator on V.
I must show that I+iT is non-singular.
I have proven that it is injective, but I have trouble with surjectivity.
My idea was to show that for each v in V there exists w in V s.t.
||w+iTw - v|| = 0, exploiting the identity (Ta|b) = (a|Tb).

Kiuhnm
.