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For <math>\mathcal{H}</math> to be a proper cone in <math>L(X; Y)</math> it is sufficient that the positive cone of ''X'' be total in ''X'' (i.e. the span of the positive cone of ''X'' be dense in ''X'').
If ''Y'' is a locally convex space of dimension greater than 0 then this condition is also necessary.{{sfn | Schaefer | 1999 | pp=225-229}}
Thus, if the positive cone of ''X'' is total in ''X'' and if ''Y'' is a locally convex space, then the canonical ordering of <math>L(X; Y)</math> defined by <math>\mathcal{H}</math> is a regular order.{{sfn | Schaefer | 1999 | pp=225-229}}
== Properties ==
Suppose that ''X'' and ''Y'' are ordered [[locally convex]] topological vector spaces (TVSs) with ''X'' being a [[Mackey space]] on which every [[positive linear functional]] is continuous. If the positive cone of ''Y'' is a [[normal cone (functional analysis)|weakly normal cone]] in ''Y'' then every positive linear operator from ''X'' into ''Y'' is continuous.{{sfn | Schaefer | 1999 | pp=225-229}}
== See also ==
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