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I am working through the Krein-Milman theorem and using Brezis "Functional Analysis and Sobolev Spaces" problem #1 as an outline. There is one issue that I am struggling with and that is the following:

Let $a\in K$, $a$ is an extreme point of $K$ iff $\{a\}$ is an extreme set.

Here $K$ is a nonempty, compact, convex subset of a normed vector space $E$. My issue is with $(\Leftarrow)$. Let $\{a\}$ be an extreme set. Let $x$ and $y$ be elements of $K$. Consider the line segment $l:tx+(1-t)y$ for $t\in[0,1]$. I do not see how to conclude that $a$ is an extreme point.

Thanks for looking.

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Suppose $M=\{a\}$ is an extreme set. Assume $a$ is not an extreme point, i.e. there exist $x,y\in K$ with $x\neq y$, $\lambda\in[0,1]$, such that $\lambda x+(1-\lambda)y=a$. Then $\lambda x+(1-\lambda)y\in M$, so $x\in M$ and $y\in M$, and thus $x=y=a$, a contradiction.

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  • $\begingroup$ That is very clear, thank you. $\endgroup$ – Joe Feb 20 '18 at 2:10
  • $\begingroup$ You're welcome. Glad to help! $\endgroup$ – Aweygan Feb 20 '18 at 2:10

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