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If ''M'' is a vector subspace of <math>\mathcal{L}(X; Y)</math> and if ''H'' ∩ ''M'' is a proper cone then this proper cone defines a '''canonical''' partial order on ''M'' making ''M'' into a partially ordered vector space.{{sfn | Schaefer | 1999 | pp=225-229}}
If (''X'', ≤) and (''Y'', ≤) are [[ordered topological vector space]]s (ordered TVSs) and if <math>\mathcal{G}</math> is a family of bounded subsets of ''X'' whose union covers ''X'' then the [[positive cone]] <math>\mathcal{H}</math> in <math>L(X; Y)</math>, which is the space of all continuous linear maps from ''X'' into ''Y'', is closed in <math>L(X; Y)</math> when <math>L(X; Y)</math> is endowed with the [[topology of uniform convergence|<math>\mathcal{G}</math>-topology]].{{sfn | Schaefer | 1999 | pp=225-229}}
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}}
== See also ==
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