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.
Ordered TVSes have important applications in spectral theory.
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 (over the real or complex numbers), 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.
- 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.