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ordered topological vector space


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Let $k$ be either $\mathbb{R}$ or $\mathbb{C}$ considered as a field.  An \emph{ordered topological vector space} $L$, (\emph{ordered t.v.s} for short) is 
\item a topological vector space over $k$, and 
\item an ordered vector space over $k$, such that 
\item 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.

\textbf{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.