Saturated family explained
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:
contains every subset of
;
- the union of any finite collection of elements of
is an element of
;
- for every scalar
contains
;
- the closed convex balanced hull of
belongs to
Definitions
If
is any collection of subsets of
then the smallest saturated family containing
is called the of
The family
is said to
if the union
} G is equal to
; it is if the linear span of this set is a dense subset of
Examples
The intersection of an arbitrary family of saturated families is a saturated family.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. 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.
