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A version of Hahn-Banach theorem says that in a real normed space, any closed convex subset is equal to the intersection of all closed half-spaces that contains it.

Is there a similar statement in the complex case ? In other words is there a notion of half-space in a complex normed space (maybe by taking the real part of a linear form ?) that will give the same result in this case?

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Yes. There is this version of Hahn-Banach:

Let $X$ be a (complex) locally convex space, and $A,B\subset X$ disjoint, nonempty, convex, with $A$ compact and $B$ closed. Then there exists a continuous linear functional $\varphi:X\to\mathbb C$ and $c,d\in\mathbb R$ with $$ \text{Re}\,\varphi(v)<c<d<\text{Re}\,\varphi(w),\ \ \ v\in A,\ \ w\in B. $$

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