4
$\begingroup$

Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no continuous extension to $\mathcal{X}$

I think this problem means that Hahn-Banach theorem ( LCS version ) may not hold in a TVS.

But I can't find a counterexample..

$\endgroup$
  • $\begingroup$ $L^p$ with $0<p<1$ $\endgroup$ – yoyo Jun 2 '13 at 12:25
  • $\begingroup$ It's a standard result that $L^p[0,1]$ has trivial dual for $0 < p < 1$. Now apply the hint I gave in my answer. $\endgroup$ – kahen Jun 2 '13 at 13:32
  • $\begingroup$ Ahh.. i understand it Thanks very much!! $\endgroup$ – functional Jun 2 '13 at 13:33
2
$\begingroup$

Hint: Do you know of a TVS with trivial dual space? Then take any non-zero linear functional on a one-dimensional subspace.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.