Cone-saturated
In mathematics, specifically in order theory and functional analysis, if is a cone at 0 in a vector space such that then a subset is said to be -saturated if where
Given a subset the -saturated hull of is the smallest -saturated subset of that contains
If is a collection of subsets of then
If is a collection of subsets of and if is a subset of then is a fundamental subfamily of if every is contained as a subset of some element of
If is a family of subsets of a TVS then a cone in is called a -cone if is a fundamental subfamily of and is a strict -cone if is a fundamental subfamily of
-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.
Properties
If is an ordered vector space with positive cone thenThe map is increasing; that is, if then
If is convex then so is When is considered as a vector field over then if is balanced then so is
If is a filter base in then the same is true of