How to prove this identity for the complex conjugate of linear operator?

Question

I want to prove the following identity:

$$\langle v|\Omega^{\dagger}|u\rangle = \langle u|\Omega | v \rangle^*$$

How should I go about this? I believe I can prove it when $\Omega$ is hermitian, but I do not know how to prove it in general.

Answer 1

By the definition. It is $$(\Omega^\dagger v, u) = (v, \Omega u) = (\Omega u, v) ^* $$

Answer 2

The complex conjugate operation distributes in an inverse order. Let a, b, c be some numbers that the conjugate action can act. Then, $$ (abc)^* = c^* b^* a^* $$ For a general case where a, b, c are not numbers but could be state, operator, etc., such that the product is a number, then one needs to use Hermitian conjugate, of course. So, $$\langle u | \Omega | v \rangle^* = | v\rangle^\dagger \Omega^\dagger \langle u |^\dagger $$ and of course, due to the fact that the conjugate of a bra is a ket and vice versa, one can conclude the proof.