0
$\begingroup$

Possible Duplicate:
Dual of a dual cone

I try to prove the following statement:

Let $V$ be a finite-dimensional ordered topological vector space ($V^{**} \cong V$) with a closed positive cone $C$. Then, then dual cone of the dual cone of C is equal to the original cone, $C^{**} = C$.

I tried to prove it as follows: Obviously, $C \subset C^{**}$. Therefore, it suffices to show that $w \notin C$ implies $w \notin C^{**}$. So let $w \in V, w \notin C$. Then, by the Hahn-Banach-Theorem, there is a linear functional $f \in V^*$ with $f(w) < \inf \{f(v) \mid v \in C\}$. What I fail to show is that $f$ can be chosen to be non-negative on $C$ (therefore lying in $C^*$ ) but negative at $w$ (therefore $w$ not lying in $C^{**}$).

I've seen that someone asked (almost) the same question, but the only answer to his post did not answer this particular question: Dual of a dual cone

Can anyone help?

  • 0
    Good, th$a$t's the kind of argument I was looking for. Thank you!2011-09-30

0 Answers 0