I have a question applying the Hahn-Banach theorem. I would apply this version of the Hahn-Banach separation theorem.

Theorem. Let $V$ be a topological vector space over $\mathbb{R}$. If $A$, $B$ are convex, non-empty disjoint subsets of $V$, then: If $A$ is open, then there exists a continuous linear map $\lambda$ : $V → K$ and $t ∈ R$ such that $λ(a) < t ≤ λ(b)$ for all $a ∈ A$, $b ∈ B$.

In this case, my question is whether we can change the constant $t$ arbitrarily. If not in general, when can we change $t$ to $0$, i.e., $λ(a) < 0 ≤ λ(b)$ for all $a ∈ A$, $b ∈ B$?

Thanks in advance.

  • 1
    $\begingroup$ You can multiply $t$ with any positive real number (multiplying $\lambda$ with the same constant). You can't generally switch signs of $t$ or choose $t = 0$, since one of the two sets can contain $0$, and then $0 \in \lambda(S)$, where $S$ is either $A$ or $B$, and if $0\in A$ or $B$ has nonempty interior, $\lambda(S)$ contains a whole neighbourhood of $0$. $\endgroup$ – Daniel Fischer Jul 27 '14 at 8:29

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