Here is the Theorem 3.4 of Rudin.

Suppose $A$ and $B$ are disjoint, nonemtpy, convex sets in a topological vector space $X$. If $A$ is open there exists $\Lambda \in X^*$ and $\gamma \in R$ s.t. $\text{Re} \Lambda x < \gamma \leq \text{Re} \Lambda y$ for every $x \in A$ and $y \in B$.

This theorem, however, concerns the whole dual space $X^*$. I wonder whether there exists theorems that guarantees the existence of separation functional in a space smaller than $X^*$. Especially, I am interested in finding a $X^{'} \subset X^*$ s.t. the above separation still holds.

Any suggestion is welcomed.


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