Lemma 5

Assume that C is weakly complete. Let $\mathscr{A}$ be the set of restrictions to C of the continuous linear forms on E. Let $(f_\lambda)_{\lambda \in \Lambda}$ be a finite family of elements of $\mathscr{A}$, and $f = \sup(f_\lambda)$. For every $x \in C$, define

*the supremum being taken over the set $S_x$ of sequences $(x_1, x_2, \ldots, x_n)$ of elements of C such that $x_1 + x_2 + \cdots + x_n = x$. Let $\mathrm{Card}\, \Lambda = p$. Then there exists $(y_1, \ldots, y_p) \in S_x$ such that $\overline{f}(x) = f(y_1) + \cdots + f(y_p)$.

Denote by $f_1, f_2, \ldots, f_p$ the elements of the family $(f_\lambda)$. For $k = 1, 2, \ldots, p$, let $C_k$ be the set of $y \in C$ such that

The $C_k$ are disjoint convex cones with union C. Let $x_1, x_2, \ldots, x_n$ in C be such that $x_1 + x_2 + \cdots + x_n = x$. Let $y_k$ be the sum of the $x_i$ that belong to $C_k$. Then $y_1 + y_2 + \cdots + y_p = x$. Since $f$ is affine on $C_k$, $f(y_1) + \cdots + f(y_p) = f(x_1) + \cdots + f(x_n)$. Therefore

where $(y_1, y_2, \ldots, y_p)$ runs over the set of sequences of $p$ points of C such that $y_1 + y_2 + \cdots + y_p = x$. Set $D = C \cap (x - C)$. Since D is compact (TVS, II, §6, No. 8, Cor. 2 of Prop. 11), so is the set of elements $(y_1, \ldots, y_p)$ of $D^p$ such that $y_1 + \cdots + y_p = x$, thus the supremum (19) is attained.