Order complete explained
In mathematics, specifically in order theory and functional analysis, a subset
of an
ordered vector space is said to be
order complete in
if for every non-empty subset
of
that is order bounded in
(meaning contained in an interval, which is a set of the form
[a,b]:=\{x\inX:a\leqxandx\leqb\},
for some
), the
supremum
' and the
infimum
both exist and are elements of
An ordered vector space is called
order complete,
Dedekind complete, a
complete vector lattice, or a
complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a
vector lattice. An ordered vector space is said to be
countably order complete if each countable subset that is bounded above has a supremum.
Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
Examples
The order dual of a vector lattice is an order complete vector lattice under its canonical ordering.
If
is a
locally convex topological vector lattice then the strong dual
is an order complete locally convex topological vector lattice under its canonical order.
Every reflexive locally convex topological vector lattice is order complete and a complete TVS.
Properties
If
is an order complete
vector lattice then for any subset
is the ordered direct sum of the band generated by
and of the band
of all elements that are disjoint from
For any subset
of
the band generated by
is
If
and
are
lattice disjoint then the band generated by
contains
and is lattice disjoint from the band generated by
which contains
