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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 $\hat{L}_{|U}=L$.

Do

Does the theorem still holds 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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Hahn-Banach theorem with real extended valued function

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 $\hat{L}_{|U}=L$.

Do the theorem still holds 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.