Krein-Milman theorem states that
If ${K}$ is a non-empty compact convex subset of a locally convex space $ {X}$, then ${\text{ext}\;K\neq\emptyset}$ and ${K=\overline{\text{co}}(\text{ext}\;K)}.$
Note that the closure is needed and here is a related question why closure is needed. Let's suppose the LCS $X$ (having predual) is equipped with weak* topology. My question is that under which condition on $K$ we will have $$ K=\mathrm{co}(\mathrm{ext}(K)).\quad(1)$$ Clearly when $K$ is a subset of finite dimensional space, then $(1)$ is true. But for the infinte case, is there any sufficient condition for $K$ to satify $(1)$? Or $(1)$ is true if and only if $K$ is a subset of finite dimensional space?
And again I am interested in the case of $X$ equipped with weak* topology. Thanks!