Saturated family

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

  1. contains every subset of ;
  2. the union of any finite collection of elements of is an element of ;
  3. for every scalar contains ;
  4. the closed convex balanced hull of belongs to [1]

Definitions

If is any collection of subsets of then the smallest saturated family containing is called the saturated hull of [1]

The family is said to cover if the union is equal to ; it is total if the linear span of this set is a dense subset of [1]

Examples

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

The set of all bounded subsets of a topological vector space is a saturated family.

See also

References

  1. ^ a b c d 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.

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