ordered topological vector space

Let $k$ be either $ℝ$ or $ℂ$ 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.