Quasi-interior point explained
In mathematics, specifically in order theory and functional analysis, an element
of an
ordered topological vector space
is called a
quasi-interior point of the positive cone
of
if
and if the order interval
[0,x]:=\{z\inZ:0\leqzandz\leqx\}
is a total subset of
; that is, if the linear span of
is a dense subset of
Properties
If
is a
separable metrizable locally convex ordered topological vector space whose positive cone
is a complete and total subset of
then the set of quasi-interior points of
is dense in
Examples
If
then a point in
is quasi-interior to the positive cone
if and only it is a weak order unit, which happens if and only if the element (which recall is an equivalence class of functions) contains a function that is
almost everywhere (with respect to
).
A point in
is quasi-interior to the positive cone
if and only if it is interior to
