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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?

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  • $\begingroup$ Apply the Hahn-Banach separation theorem (the second part) with compact set $\{x\}$ and and closed set $Y$. $\endgroup$ – Aweygan Dec 4 '17 at 21:38
  • $\begingroup$ @Aweygan Would like to write that as an answer? $\endgroup$ – punctured dusk Aug 9 '18 at 13:14
  • $\begingroup$ @barto I don't really see the point, but alright $\endgroup$ – Aweygan Aug 9 '18 at 14:16
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Apply the Hahn-Banach separation theorem, with the compact set $\{x\}$ and the closed set $Y$. We obtain some $f\in X^*$ such that $f(x)\notin f(Y)$ (actually we obtain something stronger, but not necessary here). As $Y$ is a subspace of $X$, $f(Y)$ is a proper subspace of $\mathbb F$ ($=\mathbb R$ or $\mathbb C$). Hence $f(Y)=\{0\}$, and $f(x)\neq0$.

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