Saturated family

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

1. $𝒢$ contains every subset of $G$;
2. the union of any finite collection of elements of $𝒢$ is an element of $𝒢$;
3. for every scalar $a,$ $𝒢$ contains $aG$;
4. the closed convex balanced hull of $G$ belongs to $𝒢.$Template:Sfn

Definitions

If $𝒮$ is any collection of subsets of $X$ then the smallest saturated family containing $𝒮$ is called the Template:Em of $𝒮.$Template:Sfn

The family $𝒢$ is said to Template:Em $X$ if the union $\bigcup _{G\in 𝒢}G$ is equal to $X$; it is Template:Em if the linear span of this set is a dense subset of $X.$Template:Sfn

Examples

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

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

See also

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References

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels