Let $\, M := \{ x \in H : L(x) = 0 \}$ where $H$ is a Hilbert Space and $L$ $ \in H^* : L \neq 0 $ . Prove that $ M^\perp$ is a one dimensional space.
$\textbf{Here is my attempt:}$
Fix $L \in H^*$ then we can assume that $\exists! \, w \in H : L(x) = \, \langle \, x \, , \, w \, \rangle $ and so we can rewrite $M$ as $\{ x \in H : \langle \, x \, , \, w \, \rangle = 0 \} $. From the definition of $M^\perp$ it seems to me trivial that $M^\perp = \{ w\}$.
Am i wrong somewhere? If yes, could you give me some hints?