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.