# 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 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 $0\leq x\in L$, $0\leq f(x)\in M$ also. Therefore $f(L^{+})\subseteq M^{+}$. Conversely, a linear map that is invariant under positive cones is monotone.

Title ordered topological vector space OrderedTopologicalVectorSpace 2013-03-22 17:03:23 2013-03-22 17:03:23 CWoo (3771) CWoo (3771) 4 CWoo (3771) Definition msc 06F20 msc 46A40 msc 06F30 ordered topological linear space