# Strength of $\sf ZF$+The weak topology on every Banach space is Hausdorff

Most functional analysis books prove that Zorn's Lemma$\implies$Hahn-Banach$\implies$"The weak topology on every Banach space is Hausdorff", however it is known that the Hahn-Banach theorem can be proved from the Boolean Prime Ideal theorem, which is strictly weaker than $\sf ZL$, and that $\sf ZF+HB$ is strictly weaker than $\sf ZF+BPI$.

Let $\sf WH$ be the assertion "The weak topology on every Banach space is Hausdorff", what's the relation between $\sf ZF$,$\sf ZF+WH$ and $\sf ZF+HB$?

In the above paper the authors show that the Hahn–Banach theorem is equivalent to the statement "Every Banach space has a nontrivial continuous functional". This means that if the Hahn–Banach theorem fails, there is a Banach space $X$ with a trivial [continuous] dual, and therefore the weak topology on $X$ is the indiscrete one.