Proposition 6

Let $f$ be an upper semi-continuous affine function in a compact convex set $X$, of the Hausdorff locally convex space $E$. Let $L$ be the set of restrictions to $X$ of continuous affine functions in $E$; the set $L'$ of the $h \in L$ such that $h(x) > f(x)$ for all $x \in X$, is then decreasing directed and its lower envelope is equal to $f$.

We may suppose that $X$ is non-empty. Let $u, v$ be two elements of $L$, such that $u(x) > f(x)$ and $v(x) > f(x)$ for all $x \in X$, and let $b$ be a constant that is an upper bound of $u$ and $v$. Let $U$ (resp. $V$) be the compact convex set of points $(x, t)$ of $X \times \mathbf{R}$ such that $u(x) \leq t \leq b$ (resp. $v(x) \leq t \leq b$), and let $F$ be the set of $(x, t) \in X \times \mathbf{R}$ such that $t \leq f(x)$; $F$ is convex and closed in $X \times \mathbf{R}$. The convex envelope $K$ of $U \cup V$ does not meet $F$, since $U \cup V$ is contained in the set of $(x, t) \in X \times \mathbf{R}$ such that $f(x) < t$, a set which is convex and does not meet $F$. As $K$ is compact (II, p. 14, prop. 15), we can separate $F$ strictly from $K$ by a closed hyperplane $H$ in $E \times \mathbf{R}$. For every $x \in X$, the hyperplane $H$ separates $(x, f(x))$ strictly from $(x, b)$, and therefore meets the line $\{ x \} \times \mathbf{R}$ in a single point $w(x)$; thus $H$ is the graph of a continuous affine function whose restriction $w$ to $X$ is a member of $L$, that is a lower bound for $u$ and $v$ and that satisfies the inequality $w(x) > f(x)$ for all $x \in X$. This proves that the set $L'$ is decreasing directed. Prop. 5 of II, p. 39, applied to $-f$ shows that $f$ is the lower envelope of $L'$.