Hahn-Banach theorem with real extended valued function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:42:49Z http://mathoverflow.net/feeds/question/109226 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109226/hahn-banach-theorem-with-real-extended-valued-function Hahn-Banach theorem with real extended valued function alef87 2012-10-09T13:55:26Z 2012-10-09T21:49:54Z <p>Hello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem:</p> <p>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 $\hat{L}_{|U}=L$.</p> <p>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?</p> <p>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. $\lbrace x\in V: p(x) \in \mathbb{R} \rbrace \neq \emptyset$ ?</p> <p>Thanks to everyone in advance for helping me.</p> http://mathoverflow.net/questions/109226/hahn-banach-theorem-with-real-extended-valued-function/109231#109231 Answer by Jochen Wengenroth for Hahn-Banach theorem with real extended valued function Jochen Wengenroth 2012-10-09T14:34:33Z 2012-10-09T14:34:33Z <p>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 $L(x,0)=x$.</p> <p>Then $L$ is $p$-dominated but there is no $p$-dominated linear extension.</p>