# Geometry Hahn Banach theorem for weak* topology

We have geometric Hahn Banach theorem in standard functional analysis course, saying that there exists a separating hyperplane separates two special sets in usual topology. However this theorem can be used for some special sets in weak* topology. This is the problem 9 in the functional analysis book by Prof. Brezis.

Theorem 1 Let ${E}$ be a Banach space, ${A,B \subset E^*}$ be two nonempty disjoint convex subsets. Assume ${A}$ is open in weak* topology. Then there exist some ${x \in E, x \neq 0}$ and a constant ${\alpha}$ such that the hyperplane ${\{ f \in E^* : \langle f, x \rangle_{E^* \times E} = \alpha \}}$ separates ${A}$ and ${B}$.