In the book "Functional analysis and semi-groups" by Hille and Phillips, the Krein-Milman theorem (p. 28, Theorem 2.6.4) is stated as follows:
Let $X$ be a TVS such that the topological dual $X'$ separates points (i.e. for any $x \in X$, $x \neq 0$, there is some $\phi \in X'$ with $\phi(x) \neq 0$). Then any compact, convex set $C \subseteq X$ is the closed convex hull of its extreme points.
No proof is given, only a reference to a paper of Kelley
J. L. Kelley, Note on a theorem of Krein and Milman, J. Osaka Inst. Sci. Tech. Part I 3 (1951)
which I cannot access. However other books that refer to the same paper always assume local convexity of the space $X$. Also the usual proof of Krein-Milman (for locally convex TVS) needs Hahn-Banach, which in its weakest geometric form assumes that the interior of one of the convex sets is non-empty, which the closed convex hull of $C$ does not need to be.
So my question is:
It the above theorem true? If yes, is is there any accessible reference where it is proven?