Let $k$ be either $\mathbb{R}$ or $\mathbb{C}$ considered as a field. An ordered topological vector space $L$ (ordered t.v.s for short) is

• a topological vector space over $k$ and
• an ordered vector space over $k$ such that
• the positive cone $L^+$ of $L$ is a closed subset of $L$

The last statement can be interpreted as follows: if a sequence of non-negative elements $x_i$ of $L$ converges to an element $x$ then $x$ is non-negative.

Remark. Let $L,M$ be two ordered t.v.s., and $f:L\to M$ a linear transformation that is monotone. Then if $0\le x\in L$ $0\le f(x)\in M$ also. Therefore $f(L^+)\subseteq M^+$ Conversely, a linear map that is invariant under positive cones is monotone.