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

is a closed subset of X.

[1] Ordered TVSes have important applications in spectral theory.

If C is a cone in a TVS X then C is normal if

is the neighborhood filter at the origin,

is the C-saturated hull of a subset U of X.

[2] If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:[2] and if X is a vector space over the reals then also:[2] If the topology on X is locally convex then the closure of a normal cone is a normal cone.

[2] If C is a normal cone in X and B is a bounded subset of X then

is bounded; in particular, every interval

[2] If X is Hausdorff then every normal cone in X is a proper cone.