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 TVSes 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:
 * 1) C is a normal cone.
 * 2) For every filter $$\mathcal{F}$$ in X, if $$\lim \mathcal{F} = 0$$ then $$\lim \left[ \mathcal{F} \right]_{C} = 0$$.
 * 3) 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:
 * 1) There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
 * 2) 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:
 * 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
 * 3) the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.