Ordered topological vector space explained

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 TVSes have important applications in spectral theory.

Normal cone

See main article: Normal cone (functional analysis).

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

l{U} = \left[ l{U} \right] _{C} , where l{U} is the neighborhood filter at the origin, \left[ l{U} \right] _{C} = \left\{ \left[ U \right] : U \in l{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:

1. C is a normal cone.
2. For every filter

l{F} in X, if \lim l{F} = 0 then \lim \left[ l{F} \right] _{C} = 0 .

1. There exists a neighborhood base

l{B} in X such that B \in l{B} implies \left[ B \cap C \right] _{C} \subseteq B .

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

1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
2. There exists a generating family

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

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

1. the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.