ordered topological vector space
Let $k$ be either $\mathbb{R}$ or $\u2102$ considered as a field. An ordered topological vector space $L$, (ordered t.v.s for short) is

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a topological vector space^{} over $k$, and

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an ordered vector space over $k$, such that

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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 $0\le x\in L$, $0\le f(x)\in M$ also. Therefore $f({L}^{+})\subseteq {M}^{+}$. Conversely, a linear map that is invariant^{} under positive cones is monotone.
Title  ordered topological vector space 

Canonical name  OrderedTopologicalVectorSpace 
Date of creation  20130322 17:03:23 
Last modified on  20130322 17:03:23 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06F20 
Classification  msc 46A40 
Classification  msc 06F30 
Synonym  ordered topological linear space 