Saturated family
In mathematics, specifically in functional analysis, a family [math]\displaystyle{ \mathcal{G} }[/math] of subsets a topological vector space (TVS) [math]\displaystyle{ X }[/math] is said to be saturated if [math]\displaystyle{ \mathcal{G} }[/math] contains a non-empty subset of [math]\displaystyle{ X }[/math] and if for every [math]\displaystyle{ G \in \mathcal{G}, }[/math] the following conditions all hold:
- [math]\displaystyle{ \mathcal{G} }[/math] contains every subset of [math]\displaystyle{ G }[/math];
- the union of any finite collection of elements of [math]\displaystyle{ \mathcal{G} }[/math] is an element of [math]\displaystyle{ \mathcal{G} }[/math];
- for every scalar [math]\displaystyle{ a, }[/math] [math]\displaystyle{ \mathcal{G} }[/math] contains [math]\displaystyle{ aG }[/math];
- the closed convex balanced hull of [math]\displaystyle{ G }[/math] belongs to [math]\displaystyle{ \mathcal{G}. }[/math][1]
Definitions
If [math]\displaystyle{ \mathcal{S} }[/math] is any collection of subsets of [math]\displaystyle{ X }[/math] then the smallest saturated family containing [math]\displaystyle{ \mathcal{S} }[/math] is called the saturated hull of [math]\displaystyle{ \mathcal{S}. }[/math][1]
The family [math]\displaystyle{ \mathcal{G} }[/math] is said to cover [math]\displaystyle{ X }[/math] if the union [math]\displaystyle{ \bigcup_{G \in \mathcal{G}} G }[/math] is equal to [math]\displaystyle{ X }[/math]; it is total if the linear span of this set is a dense subset of [math]\displaystyle{ X. }[/math][1]
Examples
The intersection of an arbitrary family of saturated families is a saturated family.[1] Since the power set of [math]\displaystyle{ X }[/math] is saturated, any given non-empty family [math]\displaystyle{ \mathcal{G} }[/math] of subsets of [math]\displaystyle{ X }[/math] containing at least one non-empty set, the saturated hull of [math]\displaystyle{ \mathcal{G} }[/math] is well-defined.[2] Note that a saturated family of subsets of [math]\displaystyle{ X }[/math] that covers [math]\displaystyle{ X }[/math] is a bornology on [math]\displaystyle{ X. }[/math]
The set of all bounded subsets of a topological vector space is a saturated family.
See also
References
- ↑ 1.0 1.1 1.2 1.3 Schaefer & Wolff 1999, pp. 79–82.
- ↑ 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 (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
 |  |
