Cone-saturated explained

In mathematics, specifically in order theory and functional analysis, if

C is a cone at 0 in a vector space X such that 0 \in C, then a subset S \subseteq X is said to be C -saturated if S = [S] _{C,} where [S] _{C} : = (S + C) \cap (S - C). Given a subset S \subseteq X, the C -saturated hull of S is the smallest C -saturated subset of X that contains S. If l{F} is a collection of subsets of X then \left[ l{F} \right] _{C} : = \left\{ [F] _{C} : F \in l{F} \right\}.

If

l{T} is a collection of subsets of X and if l{F} is a subset of l{T} then l{F} is a fundamental subfamily of l{T} if every T \in l{T} is contained as a subset of some element of l{F}. If l{G} is a family of subsets of a TVS X then a cone C in X is called a l{G} -cone if \left\{ \overline{[G] _{C}} : G \in l{G} \right\} is a fundamental subfamily of l{G} and C is a strict l{G} -cone if \left\{ [B] _{C} : B \in l{B} \right\} is a fundamental subfamily of l{B}. C -saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Properties

If

X is an ordered vector space with positive cone C then [S] _{C} = c up \left\{ [x, y] : x, y \in S \right\}.

The map

S \mapsto [S] _{C} is increasing; that is, if R \subseteq S then [R] _{C} \subseteq [S] _{C.} If S is convex then so is [S] _{C.} When X is considered as a vector field over \R, then if S is balanced then so is [S] _{C.}

If

l{F} is a filter base (resp. a filter) in X then the same is true of \left[ l{F} \right] _{C} : = \left\{ [ F ] _{C} : F \in l{F} \right\}.