Proposition 5

Every function $h \in \mathscr{H}$ attains its supremum at at least one $\mathscr{H}$*-extremal point*.

Let $x$ be a point of $X$, $h$ a function in $\mathscr{H}$. The relation $h(z) \leq h(x)$ for all $z \in X$ may be written $\langle h, i_{\mathscr{H}}(z) \rangle \leq \langle h, i_{\mathscr{H}}(x) \rangle$ for all $z \in X$, and therefore means that the weakly closed hyperplane of $\mathscr{H}'$ with equation $\langle h, t' \rangle = \langle h, i_{\mathscr{H}}(x) \rangle$ is a support hyperplane of $i_{\mathscr{H}}(X)$. It is known (TVS, II, §7, No. 1, Cor. of Prop. 1) that such a hyperplane contains at least one extremal point of the closed convex envelope of $i_{\mathscr{H}}(X)$, and such a point $i_{\mathscr{H}}(y)$ is the image of an $\mathscr{H}$-extremal point $y$ by definition; $h(y)$ is therefore equal to the supremum of $h$ in $X$.