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 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.

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See also

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.CS1 maint: ref=harv (link)
  • Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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