Hahn Banach theorem for locally convex topological vector space.

Question

Given a locally convex topological vector space $X$, and a closed proper subspace $Y \subset X$. Take $x \in X \setminus Y$. Is it true we can find a continuous linear functional $f : X \to \mathbb R$, such that $f(x) \neq 0$, and $f|_Y \equiv 0$. I know that this is true for normed vector spaces. However, can we do this for general topological vector spaces?