Let $V$ be a Hermitian finite-dimensional vector space and $f$ is an operator on $V$ and if $W$ is $f$-invariant then $W^{\perp}$ is $f$-invariant. Prove that $f$ is normal operator.
Proof: Let's prove by induction on $\dim V$.
If $\dim V=1$, i.e. $V=\langle e_1\rangle$ then $f(e_1)=\lambda e_1$. Then $f^*(e_1)=\mu e_1$ and it's trivial to check that $ff^*=f^*f$, i.e. $f$ is normal. Question: Am I right that in this case I do not use anything from the problem statement?
Suppose it is true for all Hermitian spaces of dimension $\leq n-1$.
Let $\dim V=n$. Since the ground field is $\mathbb{C}$ then there is eigenvector $v$ associated with eigenvalue $\lambda$. Let $W=\langle v\rangle $, then $\dim W^{\perp}=n-1$. Since $W$ is $f$-invariant then $W^{\perp}$ is $f$-invariant and I can consider restriction $h:=f|_{W^{\perp}}$ where $h:W^{\perp}\to W^{\perp}$ is operator. Our goal to apply induction hypothesis to operator $h$. But in order to do that we have to show that $h$ has the desired property: if $U$ is $h$-invariant then $U^{\perp}$ is $h$-invariant (here by $U^{\perp}$ I mean orthogonal complement relative to $W^{\perp}$).
Let $U$ is $h$-invariant subspace of $W^{\perp}$ then $h(U)=f(U)\subseteq U$ which shows that $U$ is $f$-invariant subspace of $V$ then it follows that $U^{\perp}$ is $f$-invariant, but by $U^{\perp}$ I mean orthogonal complement relative to $V$. Then I am got confused completely.
Can anyone show how to make this reasoning more clearer, please?