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 such that for all and , and since is a linear space we have for all , and considering positive/ negative, we get for all , and for all . Now consider .