Ordered Topological Vector Space

In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) ''X'' that has a partial order ≤ making it into an ordered vector space whose positive cone $C := \left\$ is a closed subset of ''X''.
Ordered TVSes have important applications in spectral theory.

Normal cone

If ''C'' is a cone in a TVS ''X'' then ''C'' is normal if \mathcal = \left \mathcal \right, where $\mathcal$ is the neighborhood filter at the origin, \left \mathcal \right = \left\, and := \left(U + C\right) \cap \left(U - C\right) is the ''C''-saturated hull of a subset ''U'' of ''X''.

If ''C'' is a cone in a TVS ''X'' (over the real or complex numbers), then the following are equivalent:
# ''C'' is a normal cone.
# For every filter $\mathcal$ in ''X'', if $\lim \mathcal = 0$ then \lim \left \mathcal \right = 0.
# There exists a neighborhood base $\mathcal$ in ''X'' such that $B \in \mathcal$ implies \left B \cap C \right \subseteq B.

and if ''X'' is a vector space over the reals then also:
# There exists a neighborhood base at the origin consisting of convex, balanced, ''C''-saturated sets.
# There exists a generating family $\mathcal$ of semi-norms on ''X'' such that $p(x) \leq p(x + y)$ for all $x, y \in C$ and $p \in \mathcal$.

If the topology on ''X'' is locally convex then the closure of a normal cone is a normal cone.

Properties

If ''C'' is a normal cone in ''X'' and ''B'' is a bounded subset of ''X'' then \left B \right is bounded; in particular, every interval $$, b/math> is bounded.
If ''X'' is Hausdorff then every normal cone in ''X'' is a proper cone.

Properties

* Let ''X'' be an ordered vector space over the reals that is finite-dimensional. Then the order of ''X'' is Archimedean if and only if the positive cone of ''X'' is closed for the unique topology under which ''X'' is a Hausdorff TVS.
* Let ''X'' be an ordered vector space over the reals with positive cone ''C''. Then the following are equivalent:
# the order of ''X'' is regular.
# ''C'' is sequentially closed for some Hausdorff locally convex TVS topology on ''X'' and $X^$ distinguishes points in ''X''
# the order of ''X'' is Archimedean and ''C'' is normal for some Hausdorff locally convex TVS topology on ''X''.

See also

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References

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{{Order theory

Functional analysis
Order theory
Topological vector spaces