Saturated family

In mathematics, specifically in functional analysis, a collection 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 the following conditions all hold:

1. for every G ∈ 𝒢, 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 G ∈ 𝒢 and scalar a, 𝒢 contains aG;
4. for every G ∈ 𝒢, the closed, convex, balanced hull of G belongs to 𝒢.[1]

If 𝒮 is any collection of subsets of X then the smallest saturated family containing 𝒮 is called the saturated hull of 𝒮.[1] 𝒢 is said to cover X if the union ${\displaystyle \cup _{G\in {\mathcal {G}}}G}$ equals X; it is total if the linear span of this set is a dense subset of X.[1]

Examples

The intersection of an arbitrary family of saturated families is a saturated family.[1] 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.[2] 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
• Vector lattice

References

1. Schaefer & Wolff 1999, pp. 79–82.
2. Schaefer & Wolff 1999, pp. 79-88.

• 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 (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.