Here is a separation theorem in $\mathbb{R}^n$: Let $A$ and $B$ be nonempty disjoint convex sets in $\mathbb{R}^n$. Then there exists a nonzero linear functional $L$ on $\mathbb{R}^n$ such that $\inf L\left(B\right)\geqslant \sup L\left(A\right)$.
I was wondering what the corresponding Hahn Banach Separation Theorem in topological vector space (or others that are more general than $\mathbb{R}^n$) is. I searched some notes online, but they require $A$ to have an interior point or to be open. I am looking for the one that only requires $A$ and $B$ to be convex and nonempty.
I realized that I can rephrase my question as: what is the infinite-dimensional version of the theorem I stated at the beginning? I suspect not...
In Brezis' functional analysis textbook, Theorem 1.7 (Hahn–Banach, second geometric form) requires that $A$ and $B$ be convex, nonempty, disjoint, with $A$ closed and $B$ compact, where $A$ and $B$ are subsets of some Banach space $E$. Then you can find a hyperplane [$f=\alpha$] strictly separates $A$ and $B$.
You may find the textbook at https://link.springer.com/book/10.1007/978-0-387-70914-7.