Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$.

If $M$ is closed, then $M=\overline{M}$. And we don't have to extend $f$. So how to extend $f$ to $\overline{M}$ if $M$ is not closed?

Thank you in advance!


closed as off-topic by Theoretical Economist, Xander Henderson, Leucippus, max_zorn, Ben Sep 5 '18 at 8:14

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  • $\begingroup$ Is this not just Hahn-Banach? $\endgroup$ – Theoretical Economist Sep 4 '18 at 20:32
  • $\begingroup$ In fact, you can use the analytic form of the Hahn-Banach theorem to obtain an extension $F \colon X \rightarrow \mathbb{K}$ of $f \colon M \rightarrow \mathbb{K}$ on all of $X$. $\endgroup$ – p4sch Sep 4 '18 at 20:37
  • $\begingroup$ @p4sch Thanks for your reply! But this is no sublinear functional in this case. $\endgroup$ – Answer Lee Sep 4 '18 at 20:42
  • $\begingroup$ @AnswerLee there is. You can use Hahn-Banach. Try and think about it for a bit — if you’re still stuck after a while I can give you another hint. $\endgroup$ – Theoretical Economist Sep 4 '18 at 20:45
  • $\begingroup$ @TheoreticalEconomist Thanks for your reply. I am sorry that I am a little slow. Can you give me some hints please! $\endgroup$ – Answer Lee Sep 4 '18 at 20:59