Saturated family

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 ${𝒢}.$

Definitions

If ${𝒮}$ is any collection of subsets of $X$ then the smallest saturated family containing ${𝒮}$ is called the saturated hull of ${𝒮}.$

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

Examples

The intersection of an arbitrary family of saturated families is a saturated family. 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. 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

• Topology of uniform convergence
• Topological vector lattice

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.