Let $X$ be a locally convex space. Let $M$ be a linear subspace of a locally convex space $X$. Let $f\in M^*$. Then can we find a seminorm $p$ on $X$ such that $|f(x)|\leq p(x)$ for all $x\in M$?

Thank you in advance!

  • $\begingroup$ Isn't $|f|$ a seminorm? $\endgroup$ – Lord Shark the Unknown Sep 6 '18 at 18:18
  • $\begingroup$ @LordSharktheUnknown Yes but it is not on $X$. $\endgroup$ – Answer Lee Sep 6 '18 at 18:18
  • $\begingroup$ Then extend $f$ it to $X$. $\endgroup$ – Lord Shark the Unknown Sep 6 '18 at 18:21
  • $\begingroup$ @LordSharktheUnknown But how? $\endgroup$ – Answer Lee Sep 6 '18 at 18:24
  • $\begingroup$ Any way! As you state the problem, you don't need any sort of continuity. $\endgroup$ – Lord Shark the Unknown Sep 6 '18 at 18:26

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