Prove Inner product is real

Question

Let $T : V \longrightarrow V ; V:= \mathbb{C^0}$ be a linear operator equiped with the inner product $<u,v>$.

Prove that $<T(u),v>$ is real.

I'm able to prove this statement if it is a self-adjoint operator; how do I prove it, if it is not self-adjoint.

Answer 1

Write:

$$T = \frac{1}{2}(T + T^*) + \frac{i}{2i}(T - T^*)$$

Prove that $$K=\frac{1}{2}(T + T^*)$$ $$\Lambda=\frac{1}{-2i}(T^* - T)$$ are self adjoint operators and go on...