Saturated family

Concept in functional analysis

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

1. ${\displaystyle {\mathcal {G}}}$ contains every subset of ${\displaystyle G}$;
2. the union of any finite collection of elements of ${\displaystyle {\mathcal {G}}}$ is an element of ${\displaystyle {\mathcal {G}}}$;
3. for every scalar ${\displaystyle a,}$ ${\displaystyle {\mathcal {G}}}$ contains ${\displaystyle aG}$;
4. the closed convex balanced hull of ${\displaystyle G}$ belongs to ${\displaystyle {\mathcal {G}}.}$^[1]

Definitions

If ${\displaystyle {\mathcal {S}}}$ is any collection of subsets of ${\displaystyle X}$ then the smallest saturated family containing ${\displaystyle {\mathcal {S}}}$ is called the saturated hull of ${\displaystyle {\mathcal {S}}.}$^[1]

The family ${\displaystyle {\mathcal {G}}}$ is said to cover ${\displaystyle X}$ if the union ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ is equal to ${\displaystyle X}$; it is total if the linear span of this set is a dense subset of ${\displaystyle X.}$^[1]

Examples

The intersection of an arbitrary family of saturated families is a saturated family.^[1]Since the power set of ${\displaystyle X}$ is saturated, any given non-empty family ${\displaystyle {\mathcal {G}}}$ of subsets of ${\displaystyle X}$ containing at least one non-empty set, the saturated hull of ${\displaystyle {\mathcal {G}}}$ is well-defined.^[2] Note that a saturated family of subsets of ${\displaystyle X}$ that covers ${\displaystyle X}$ is a bornology on ${\displaystyle X.}$

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 – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

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. Vol. 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.