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 formula_1 is a closed subset of "X". Ordered TVS have important applications in spectral theory. Normal cone. If "C" is a cone in a TVS "X" then "C" is normal if formula_2, where formula_3 is the neighborhood filter at the origin, formula_4, and formula_5 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: and if "X" is a vector space over the reals then also: 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 formula_16 is bounded; in particular, every interval formula_17 is bounded. If "X" is Hausdorff then every normal cone in "X" is a proper cone.