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

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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 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
Canonical name	OrderedTopologicalVectorSpace
Date of creation	2013-03-22 17:03:23
Last modified on	2013-03-22 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