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I have such a problem to solve:

Let $K$ and $L$ be two non-empty, convex sets in a normed space $X$, such that $\inf\left\{\Vert x-y\Vert : \ x\in K, y\in L\right\}>0$. Prove, that there exists non-zero, continuous and linear functional $\varphi: X\rightarrow \mathbb{K}$ such that: $$\sup\left\{\Re (\varphi(x)): x\in K\right\}\le\inf\left\{\Re (\varphi(x)): x\in L\right\} \ \ \ \ \ \ \ \ \ \ (*) $$

Every normed space is a topological vector space and the condition from assupmtions means, that $K$ and $L$ are disjoint. For me it looks like direct consequence of the Hahn-Banach separation theorem. I'm confusing with the fact that $K$ has to be open. How can I show this? Or maybe I have to use another tool.

There is also a second question: due to above assumptions, can we always pick such $\varphi$ that in $(*)$ we'll get a strict inequality?

Any hint to those questions?

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