Example

\* Let H be a real Hilbert space; in the vector space $\mathscr{L}(H)$ of continuous endomorphisms of H, the positive hermitian endomorphisms form a proper pointed convex cone; this cone, therefore, defines an order structure compatible with the vector space structure of $\mathscr{L}(H)$ and for which the relation $A \leqslant B$ means that $B - A$ is a positive hermitian endomorphism. \*

For any pointed convex cone P in the vector space E, the set $P \cap (-P)$ is a vector subspace, H, of E (II, p. 11, cor. 2). The canonical image $P'$ of P in $E/H$ is a convex cone and the inverse image of $P'$ in E is P. Thus $P' \cap (-P') = \{0\}$, and $P'$ defines an order structure on $E/H$ that is compatible with its vector space structure.

A linear form $f$ on a preordered vector space E is said to be positive if $x \geqslant 0$ in E implies $f(x) \geqslant 0$. Or, alternatively, if the convex cone P of elements $\geqslant 0$ in E is contained in the half space of those $x$ for which $f(x) \geqslant 0$. Clearly, in the dual $E^*$ to E, the set of positive linear forms is a pointed convex cone.