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saturated family

{{Short description|Concept in functional analysis}}

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

  1. \mathcal{G} contains every subset of G;
  2. the union of any finite collection of elements of \mathcal{G} is an element of \mathcal{G};
  3. for every scalar a, \mathcal{G} contains aG;
  4. the closed convex balanced hull of G belongs to \mathcal{G}.{{sfn|Schaefer|Wolff|1999|pp=79–82}}

Definitions

If \mathcal{S} is any collection of subsets of X then the smallest saturated family containing \mathcal{S} is called the {{em|saturated hull}} of \mathcal{S}.{{sfn|Schaefer|Wolff|1999|pp=79–82}}

The family \mathcal{G} is said to {{em|cover}} X if the union \bigcup_{G \in \mathcal{G}} G is equal to X;

it is {{em|total}} if the linear span of this set is a dense subset of X.{{sfn|Schaefer|Wolff|1999|pp=79–82}}

Examples

The intersection of an arbitrary family of saturated families is a saturated family.{{sfn|Schaefer|Wolff|1999|pp=79–82}}

Since the power set of X is saturated, any given non-empty family \mathcal{G} of subsets of X containing at least one non-empty set, the saturated hull of \mathcal{G} is well-defined.{{sfn|Schaefer|Wolff|1999|pp=79-88}}

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

  • {{annotated link|Topology of uniform convergence}}
  • {{annotated link|Topological vector lattice}}
  • {{annotated link|Vector lattice}}

References

{{reflist}}

  • {{Narici Beckenstein Topological Vector Spaces|edition=2}}
  • {{Schaefer Wolff Topological Vector Spaces|edition=2}}
  • {{Trèves François Topological vector spaces, distributions and kernels}}

{{Duality and spaces of linear maps}}

{{Topological vector spaces}}

{{Boundedness and bornology}}

{{Functional analysis}}

Category:Functional analysis