Saturated family

In mathematics, specifically in functional analysis, a family of subsets a topological vector space (TVS) is said to be saturated if contains a non-empty subset of and if for every the following conditions all hold:

1. contains every subset of ;
2. the union of any finite collection of elements of is an element of ;
3. for every scalar contains ;
4. the closed convex balanced hull of belongs to

Definitions

If is any collection of subsets of then the smallest saturated family containing is called the of

The family is said to if the union is equal to ; it is if the linear span of this set is a dense subset of

Examples

The intersection of an arbitrary family of saturated families is a saturated family. Since the power set of is saturated, any given non-empty family of subsets of containing at least one non-empty set, the saturated hull of is well-defined. Note that a saturated family of subsets of that covers is a bornology on

The set of all bounded subsets of a topological vector space is a saturated family.

See also

References