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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..

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$L^p$ with $0<p<1$ – yoyo Jun 2 '13 at 12:25
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. – kahen Jun 2 '13 at 13:32
Ahh.. i understand it Thanks very much!! – functional Jun 2 '13 at 13:33
up vote 2 down vote accepted

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

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