Let $X$ be an inner product space, and define a subspace of it as $H=\{x \in X : \langle x,v \rangle =0\}$ for some $v \in X$. I was wondering if it was true that any vector, say $y$, orthogonal to $H$ is collinear with $v$. In other terms, $\langle x,v \rangle =0$ and $\langle x,y \rangle =0$ $ \forall x \in H$ $\iff y = \alpha v$ for some $\alpha \in \mathbb{R}$.
I know that the set of vectors orthogonal to $H$ is a subspace itself, is the relation I have in mind also true? If it is, how can I prove that?