Lemma 1

Let $E$ be a Hausdorff weak space and $C$ a proper cone with vertex 0 in $E$, that is complete for the uniform structure induced by that of $E$. Every continuous linear form in $E$ is then the difference between two continuous linear forms in $E$ that are positive in $C$.

Let $E'$ be the dual of $E$ and $F$ be the algebraic dual of $E'$, with the topology $\sigma(F, E')$. Let $H = C^\circ - C^\circ$ be the vector subspace of $E'$ formed by the differences of linear forms that are continuous in $E$ and positive in $C$ (II, p. 44, prop. 4). It is sufficient to show that the orthogonal to $H$ in $F$ is $\{0\}$ (II, p. 41, Example 1). Then let $a \in F$ be orthogonal to $H$; as $a$ is orthogonal to $C^\circ$, it must belong to the bipolar of $C$ in $F$. But $E$ is identifiable as a subspace of $F$, and since $C$ is complete, thus closed in $F$, we have $a \in C$ (II, p. 44, th. 1). Similarly $a$ is orthogonal to $-C^\circ$ and therefore $a \in -C$. As $C$ is proper, we have $a = 0$.