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 $C:=\{x\in X:x\ge 0\}$ is a closed subset of X.Template:Sfn Ordered TVSes have important applications in spectral theory.

Normal cone

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

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:Template:Sfn

1. C is a normal cone.
2. For every filter $ℱ$ in X, if $\lim ℱ=0$ then $\lim {\left[ℱ\right]}_C=0$.
3. There exists a neighborhood base $ℬ$ in X such that $B\in ℬ$ implies ${[B\cap C]}_C\subseteq B$.

and if X is a vector space over the reals then also:Template:Sfn

1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
2. There exists a generating family $𝒫$ of semi-norms on X such that $p(x)\le p(x+y)$ for all $x,y\in C$ and $p\in 𝒫$.

If the topology on X is locally convex then the closure of a normal cone is a normal cone.Template:Sfn

Properties

If C is a normal cone in X and B is a bounded subset of X then ${\left[B\right]}_C$ is bounded; in particular, every interval $[a,b]$ is bounded.Template:Sfn If X is Hausdorff then every normal cone in X is a proper cone.Template:Sfn

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.Template:Sfn
• Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:Template:Sfn

1. the order of X is regular.
2. C is sequentially closed for some Hausdorff locally convex TVS topology on X and $X^{+}$ distinguishes points in X
3. 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

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces