Proposition 11

Let $E$ be a Hausdorff weak space, and $C$ be a proper convex cone with vertex 0 in $E$ and which is complete in the uniform structure induced by that of $E$.

Then there exists a set I and a continuous linear mapping u of E in the product space $\mathbf{R}^I$ with the following properties:

a) u is an isomorphism of C on $u(C)$ for the uniform structures induced respectively by those of E and of $\mathbf{R}^I$.

b) We have $u(C) \subset \mathbf{R}_+^I$.

Further, if the uniform structure induced on C by that of E is metrisable, then we can take $I = \mathbf{N}$.

Let $(f_i)_{i \in I}$ be a family of continuous linear forms in E such that the finite sums of pseudometrics of the form $(x, y) \mapsto |f_i(x - y)|$ on $C \times C$ define the uniform structure of C. (If the structure is metrisable we can take $I = \mathbf{N}$.) By lemma 1 we can suppose further that each of the $f_i$ is positive in C. Let u be the linear mapping $x \mapsto (f_i(x))_{i \in I}$ of E in $\mathbf{R}^I$. It is clear that u is continuous and that $u(C) \subset \mathbf{R}_+^I$. The restriction $u|C$ is a uniformly continuous mapping that is surjective from C on $u(C)$. Further if $x, y$ in C are such that $f_i(x) = f_i(y)$ for all $i \in I$, then $x = y$ since the uniform structure of C is Hausdorff; therefore $u|C$ is bijective. Finally, if W is an entourage of the uniform structure of C, then there exists a finite set J of I and a number $\varepsilon > 0$ such that the relations $|f_i(x) - f_i(y)| \leq \varepsilon$ for $i \in J$ imply $(x, y) \in W$; therefore $u|C$ is an isomorphism of C on $u(C)$ for the uniform structures being considered.