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.

Re: how to modify Hahn-Banach Seperation Theorem?

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