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[parent] ordered topological vector space (Definition)

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

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\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.



"ordered topological vector space" is owned by CWoo.
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Other names:  ordered topological linear space

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Cross-references: invariant, monotone, linear transformation, converges, sequence, closed subset, positive cone, ordered vector space, topological vector space, field
There are 3 references to this entry.

This is version 1 of ordered topological vector space, born on 2007-05-08.
Object id is 9347, canonical name is OrderedTopologicalVectorSpace.
Accessed 810 times total.

Classification:
AMS MSC: 06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
 46A40 (Functional analysis :: Topological linear spaces and related structures :: Ordered topological linear spaces, vector lattices)
 06F30 (Order, lattices, ordered algebraic structures :: Ordered structures :: Topological lattices, order topologies)
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