When studying the Hahn-Banach theorem, one can demonstrate the geometric version from scratch and use it to prove the analytic version, as is outlined in Hahn-Banach theorem: 2 versions.

To do so, it is necessary to show that the cone $ C = \{(x,t)\,:\,p(x) \lt t\} \subset X $ is open. ¿How is this done?


The topology you use on $C $ is the product topology, so basic open sets are products of an open set in $E $ and an open set in $\mathbb R $. We can write $$ C=\bigcup_{t>0}\{x:\ p (x)<t\}\times (-\infty,t). $$

  • $\begingroup$ And how do we know $ \{x:\ p (x)<t\} $ is open? p is supposed to be semilinear but I'm not sure if that implies continuity $\endgroup$ – Emilio Jul 29 '15 at 18:02
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    $\begingroup$ Continuity with respect to what? The topology on $E$ is the one given by $p$, so it is generated by the sets $\{x:\ p(x-y)<r\}$. $\endgroup$ – Martin Argerami Jul 29 '15 at 18:46
  • $\begingroup$ I'm considering $E$ with a previously defined topology, given by a norm. Now I start to think that this proof doesn't apply to this situation $\endgroup$ – Emilio Jul 29 '15 at 18:53
  • $\begingroup$ That's your choice. But then you are not following the proof in the answer you quoted. And what's your $p$, if it is not related to the topology? $\endgroup$ – Martin Argerami Jul 29 '15 at 18:55
  • $\begingroup$ In my version of the geometric version (from the book by Brezis) $p$ is the Minkowsky functional and the space is normed. I wanted a similar setting to go back. That's because I say that now I see there are different situations $\endgroup$ – Emilio Jul 29 '15 at 19:56

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