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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 [1] 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 [1]

-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 then [1]

The 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 [1]

If is a filter base (resp. a filter) in then the same is true of

See also

References

  1. ^ a b c d Schaefer & Wolff 1999, pp. 215–222.

Bibliography