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In mathematics, specifically in [[Functional analysis|functional analysis]], a [[Family of sets|family]] <math>\mathcal{G}</math> of subsets a [[Topological vector space|topological vector space]] (TVS) <math>X</math> is said to be '''saturated''' if <math>\mathcal{G}</math> contains a non-empty subset of <math>X</math> and if for every <math>G \in \mathcal{G},</math> the following conditions all hold:
# <math>\mathcal{G}</math> contains every subset of <math>G</math>; 
# the union of any finite collection of elements of <math>\mathcal{G}</math> is an element of <math>\mathcal{G}</math>; 
# for every scalar <math>a,</math> <math>\mathcal{G}</math> contains <math>aG</math>;
# the closed convex balanced hull of <math>G</math> belongs to <math>\mathcal{G}.</math>{{sfn|Schaefer|Wolff|1999|pp=79–82}}

==Definitions==

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

The family <math>\mathcal{G}</math> is said to {{em|cover}} <math>X</math> if the union <math>\bigcup_{G \in \mathcal{G}} G</math> is equal to <math>X</math>; 
it is {{em|total}} if the linear span of this set is a dense subset of <math>X.</math>{{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|power set]] of <math>X</math> is saturated, any given non-empty family <math>\mathcal{G}</math> of subsets of <math>X</math> containing at least one non-empty set, the saturated hull of <math>\mathcal{G}</math> is well-defined.{{sfn|Schaefer|Wolff|1999|pp=79-88}} 
Note that a saturated family of subsets of <math>X</math> that covers <math>X</math> is a [[Bornology|bornology]] on <math>X.</math>

The set of all bounded subsets of a [[Topological vector space|topological vector space]] is a saturated family.

==See also==

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

==References==

{{reflist}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolf|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->

{{Functional analysis}}

[[Category:Functional analysis]]

{{Sourceattribution|Saturated family}}