Positive linear operator: Difference between revisions

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== Properties ==

:'''Proposition''':{{sfn | Schaefer | 1999 | pp=225-229}}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}}

:'''Proposition''':{{sfn | Schaefer | 1999 | pp=225-229}} Suppose ''X'' is a [[barreled space|barreled]] [[ordered topological vector space]] (TVS) with positive cone ''C'' that satisfies ''X'' = ''C'' - ''C'' and ''Y'' is a [[semi-reflexive]] ordered TVS with a positive cone ''D'' that is a [[normal cone (functional analysis)|normal cone]]. Give ''L''(''X''; ''Y'') its canonical order and let <math>\mathcal{U}</math> be a subset of ''L''(''X''; ''Y'') that is directed upward and either majorized (i.e. bounded above by some element of ''L''(''X''; ''Y'')) or simply bounded. Then <math>u = \sup \mathcal{U}</math> exists and the section filter <math>\mathcal{F}\left( \mathcal{U} \right)</math> converges to ''u'' uniformly on every precompact subset of ''X''.

== See also ==