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Hello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem:

Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any convex function. Let $W$ be a vector subspace of $V$ and let $L:W\rightarrow \mathbb{R}$ be a linear functional dominated by $p$ on $W$. Then there is a (not generally unique) linear extension $\hat{L}$ of $L$ to $V$ that is dominated by $p$ on $V$. Furthermore $\hat{L}_{|U}=L$.

Does the theorem still hold when $p:V\rightarrow(-\infty,+\infty]$ ? Is someone able to give me a proof or to provide a counter-example that show that the theorem does not hold? And if it does not hold, it is possible to add some conditions that make it still true?

To put it in another way, is the same true if we further relax the hypothesis on $p$ and allow it to be real extended with nontrivial domain, i.e. $\lbrace x\in V: p(x) \in \mathbb{R} \rbrace \neq \emptyset $ ?

Thanks to everyone in advance for helping me.

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  • $\begingroup$ "And if it does not hold, it is possible to add some conditions that make it still true?" -- of course it is. But it would help if you thought more, and then told us, what kind of extra conditions you are prepared to impose. $\endgroup$
    – Yemon Choi
    Commented Oct 9, 2012 at 22:27

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The Hahn-Banach theorem is wrong for extended real $p$: For $V=\mathbb R^2$ let $p$ be the Minkowski functional of $A= \mathbb R \times (0,\infty)$ (so that $p(x,y)=0$ if $y>0$ and $p(x,y)=\infty$ if $y\le 0$), $W= \mathbb R \times \lbrace 0 \rbrace$, and $L: W\to\mathbb R$ defined by $L(x,0)=x$.

Then $L$ is $p$-dominated but there is no $p$-dominated linear extension.

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  • $\begingroup$ Thank you very much for the counter-example. Have you any idea of additional hypothesis that would make the theorem still true with this kind of function? $\endgroup$
    – alef87
    Commented Oct 9, 2012 at 21:48
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    $\begingroup$ I think that if p is convex, positively homogeneous and lower semicontinuous, then Hahn-Banach would hold true. $\endgroup$ Commented Oct 9, 2012 at 23:29
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    $\begingroup$ @LuisSilvestre: that sounds right, if you take the lower semicontinuity requirement to refer to the finest locally convex topology. Do you have a reference for that version of Hahn-Banach? I'm currently writing a paper which uses this and would like to attribute it appropriately. $\endgroup$ Commented Nov 29, 2014 at 2:39

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