Let $V$ be complex inner product space and let $T$ be mapping from $V$ to $V$. Prove that if the subspace $W$ is $T$-invariant, then $W^\perp$ is $T^*$-invariant.
I would like to ask if my proof is alright. My proof:
If $W$ is $T$-invariant then we have that $T(w) \in W, \forall w \in W$, hence for $w \in W^\perp$: $$\langle T(v), w\rangle = 0$$ $$\langle v, T^*(w) \rangle = 0, \forall v \in V$$ $\Rightarrow T^*(w) \in W^\perp \Rightarrow W^\perp$ is $T^*$-invariant. $\blacksquare$
I wonder if this is correct, and the fact that I did not use that V was a complex inner product is kind of worrying. Thanks a lot!