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ordered topological vector space
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 nonnegative elements $x_{i}$ of $L$ converges to an element $x$, then $x$ is nonnegative.
Remark. Let $L,M$ be two ordered t.v.s., and $f:L\to M$ a linear transformation that is monotone. Then if $0\leq x\in L$, $0\leq f(x)\in M$ also. Therefore $f(L^{+})\subseteq M^{+}$. Conversely, a linear map that is invariant under positive cones is monotone.
Synonym:
ordered topological linear space
Type of Math Object:
Definition
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