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] X that has a partial order ≤ making it into an ordered vector space whose positive cone 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, where is the neighborhood filter at the origin,, and is the C-saturated hull of a subset U of X.
If C is a cone in a TVS X, then the following are equivalent:

C is a normal cone.
For every filter in X, if then.
There exists a neighborhood base in X such that implies.

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 of semi-norms on X such that for all and.

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 is bounded; in particular, every interval 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 distinguishes points in X
the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.