Exercise 1, § 3

Let P be a proper pointed convex cone, with vertex 0, in E a vector space over $\mathbf{R}$ and p be a semi-norm on E and V the set of points $x \in E$ for which $p(x) < 1$. Let M be a vector subspace of E and f be a linear form on M. There exists a linear form g on E, which extends f and is such that it is $\geq 0$ in P and $|g(x)| \leq p(x)$ for all $x \in E$, if and only if, for all $x \in M \cap (V + P)$, we have $f(x) > -1$. (To see that the condition is sufficient consider a point $x_0 \in M$ such that, $f(x_0) = 1$, the cone Q of vertex 0 generated by $x_0 + V$, and apply the cor. of the prop. 1 (II, p. 21) to the space E carrying the relation of preorder for which $P + Q$ is the cone of elements $\geq 0$.)