Lemma 4.3.3.label Let $E$ be a locally convex space over $\R$, $K\suf E$ be nonempty and compact, and $\phi\in E^{*}$. Let $\al=\SUP{x\in K}\inner{x}{\phi}_{E}$, then $A=\curl{\phi=\al}\cap K$ is a non-empty extreme subset of $K$.
Proof. Since $K$ is compact, $\al<\infty$ and $A$ is non-empty by Proposition 3.10.3. Let $x\in A$ and $y,z\in K$ such that $x\in (y,z)$. Since $K$ is compact $\inner{y}{\phi}_{E}=\inner{z}{\phi}_{E}=\al$ thus $y,z\in A$ as well.$\square$
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