Proposition 2

Let E be a real vector space, C a convex cone with vertex 0 in E such that $E = C - C$, and $x \mapsto M(x)$ a mapping of C into $\mathbf{R}$ such that $M(\lambda x + \mu y) = \lambda M(x) + \mu M(y)$ for all $x \in C$, $y \in C$, $\lambda \geq 0$, $\mu \geq 0$. Then, there exists one and only one linear form L that extends M to E.

By hypothesis, every $z \in E$ may be written $z = y - x$, where $x, y$ belong to C; moreover, if also $z = y' - x'$ with $x' \in C$, $y' \in C$, then

$M(y) - M(x) = M(y') - M(x')$; for, from the relation $y - x = y' - x'$ we infer $y + x' = x + y'$, consequently $M(y) + M(x') = M(x) + M(y')$. Let us denote by $L(z)$ the common value of $M(y) - M(x)$ for all expressions of $z$ as the difference $y - x$ of two elements of $C$; one verifies immediately that $L$ is a linear form on $E$ extending $M$; the uniqueness of $L$ results from the fact that $C$ generates the space $E$.