Where can I find the proof of this version of Hahn-Banach theorem :

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 $\varphi :X\rightarrow \mathbb C$ and $c,d\in\mathbb R$ with

$Re\varphi(w) \gt c \gt d \gt Re\varphi(v), v\in A, w\in B.$

  • $\begingroup$ Hint: A complex vector space is, in particular, a real vector space. Given a real-linear functional $\psi$, there exists a complex-linear $\phi$ such that $\psi(w) = \mathrm{Re}\;\phi(w)$ for all $w$. From the formula you get when you do this, you see that if $\psi$ is continuous then so is $\phi$. $\endgroup$ – GEdgar Apr 12 '18 at 17:14
  • $\begingroup$ thanks, I get it $\endgroup$ – Ayman Aoufi Apr 12 '18 at 17:44

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