i was trying to solve this exercise from Royden:

Let X be a locally convex topological vector space, let $Y \subset X$ be a closed subspace and $x_0 \in X-Y$.Prove that there exists a continuous linear functional $\varphi: X \rightarrow \mathbb{R}$ such that $\varphi(y) = 0$ if $y \in Y$ and $\varphi(x_0) \ne 0$

I don't understand where to use the fact that $Y$ is closed. My attempt was to construct a positive homogeneous and subadditive functional $p_y(x) = 1 - \chi_{Y}$. It's not hard to see that it has the desired properties. Moreover it is clear that $p_y(x_0) = 1$. Thus one can take $\varphi(\lambda x_0) = \lambda$ and extend it to a linear functional $\varphi \le p_y$ on $X$. Observing that $p_y(x) \le 1$ we have that $\varphi$ is bounded on $X$ and thus, because $X$ is locally convex topological vector spaces, continuous.

Isn't it right?

  • $\begingroup$ What formulation of the hahn banach theorem are you using? I only ask because your function $p_y$ is not sublinear. $\endgroup$ – Aweygan Dec 28 '16 at 15:12
  • $\begingroup$ Why not? Let $x,y \in X$ if $x+y \in Y$ it's obvious since $p_y(x+y) =0$ otherwise, since $Y$ is a subspace of X, we must have $x \not\in Y$ or $y \not\in X$. Thus subadditivity is true also in this case. $\endgroup$ – jJjjJ Dec 28 '16 at 18:27
  • $\begingroup$ I forgot to tag @Aweygan $\endgroup$ – jJjjJ Dec 28 '16 at 20:11
  • $\begingroup$ $p_y$ is just the indicator function for $X\setminus Y$. Indicator functions are not positive homogeneous. $\endgroup$ – Aweygan Dec 28 '16 at 20:16
  • $\begingroup$ @Aweygan right. Sorry, was stupid $\endgroup$ – jJjjJ Dec 28 '16 at 20:19

Since both the sets $Y$ and $\{x_0 \}$ are closed and $\{x_0\}$ is compact then from separation theorem there exists a linear functional $f: X\to \mathbb{R} $ and real numbers $a< b$ such that $f(y)<a $ for $y\in Y $ and $f(x_0 ) =b .$ Now, take any $u\in Y$ then $$f(ny )<a $$ for all $n\in \mathbb{N}$ hence $f(y)\leqslant 0$ and analogously $$f(-ny )<a$$ thus $f(y) \geqslant 0$ and finally $$f(y)=0.$$

  • $\begingroup$ Yes that's ok, but I asked if my solution is correct or not, since I don't use the fact that $Y$ is closed $\endgroup$ – jJjjJ Dec 27 '16 at 17:14

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