Corollary

In a topological vector space $E$ with a compatible preorder structure, let $P$ be the set of elements $\geq 0$ in $E$. Let $V$ be a vector subspace of $E$ containing at least one interior point $x_0$ of $P$. Then every positive linear form on $V$ can be extended to a positive linear form on $E$.

By prop. 1 it is sufficient to show that for every $x \in E$, there exists $x' \in V$ such that $x' - x \in P$. Now let $U$ be a neighbourhood of $0$ in $E$ such that $x_0 + U \subset P$. Then $x + x_0 + U \subset x + P$, and, hence there exists $\varepsilon$ such that $0 < \varepsilon < 1$ and the point $y = x_0 + (1 - \varepsilon)x$ belongs to $x + P$; then every point of the form $x + \lambda(y - x)$ belongs to $x + P$ for $\lambda > 0$. If we take $\lambda = 1/\varepsilon$, then $x + \lambda(y - x) = \lambda x_0 \in V$, from which the conclusion follows.

The conclusion of the corollary is not necessarily valid if one does not assume that $V$ contains an interior point of $P$, even if $E$ is of finite dimension and if $P \cap V$ contains points interior in $V$ (II, p. 91, exerc. 25, b)).