Math Help - how to modify Hahn-Banach Seperation Theorem?

1. how to modify Hahn-Banach Seperation Theorem?

Suppose X is a real normed space, B is a non-empty open convex subset of X, E is a linear subspace of X, B don't intersect with E.
Then there exists a linear functional f, such that f(x)=0 when x belong to E, and f(x)>0 when x belong to B.
i don't know how to use Hahn-Banach Seperation Theorem to get this.Who can help me?Thx a lot.

2. Re: how to modify Hahn-Banach Seperation Theorem?

We can find a linear functional $g$ such that $g(x) for all $x\in B$ and $y\in E$, and since $E$ is a linear space we have for all $\alpha\in\mathbb R, g(x)<\alpha g(y)$, and considering $\alpha$ positive/ negative, we get $g(y)=0$ for all $y\in E$, and $g(x)<0$ for all $x \in B$. Now consider $f:=-g$.