Ordered topological vector space
# 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 $C:=\left\{x\in X:x\geq 0\right\}$ is a closed subset of X. Ordered TVS have important applications in spectral theory.

## Normal cone

If C is a cone in a TVS X then C is normal if ${\mathcal {U}}=\left[{\mathcal {U}}\right]_{C}$, where ${\mathcal {U}}$ is the neighborhood filter at the origin, $\left[{\mathcal {U}}\right]_{C}=\left\{\left[U\right]:U\in {\mathcal {U}}\right\}$, and $[U]_{C}:=\left(U+C\right)\cap \left(U-C\right)$ 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 ${\mathcal {F}}$ in X, if $\lim {\mathcal {F}}=0$ then $\lim \left[{\mathcal {F}}\right]_{C}=0$.
* There exists a neighborhood base ${\mathcal {B}}$ in X such that $B\in {\mathcal {B}}$ implies $\left[B\cap C\right]_{C}\subseteq B$.

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 ${\mathcal {P}}$ of semi-norms on X such that $p(x)\leq p(x+y)$ for all $x,y\in C$ and $p\in {\mathcal {P}}$.

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 $\left[B\right]_{C}$ is bounded; in particular, every interval $[a,b]$ 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.
