Content deleted Content added
Added info |
| Schaefer | |
||
Line 9:
Throughout let (''X'', ≤) and (''Y'', ≤) be preordered vector spaces on ''X'' and let <math>\mathcal{L}(X; Y)</math> be the space of all linear maps from ''X'' into ''Y''.
The set ''H'' of all positive linear operators in <math>\mathcal{L}(X; Y)</math> is a cone in <math>\mathcal{L}(X; Y)</math> that defines a preorder on <math>\mathcal{L}(X; Y)</math>.
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 | Wolff | 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 | Wolff | 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 | Wolff | 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 | Wolff | 1999 | pp=225-229}}
== Properties ==
:'''Proposition''':{{sfn | Schaefer | Wolff | 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.
:'''Proposition''':{{sfn | Schaefer | Wolff | 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 ==
Line 31:
{{Reflist}}
* {{cite book | last=Schaefer | first=Helmut H. | authorlink=Helmut H. Schaefer | last2=Wolff | first2=Manfred P. | title=Topological Vector Spaces | publisher=Springer New York Imprint Springer | series=[[Graduate Texts in Mathematics|GTM]] | volume={{{volume| 3 }}} | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7155-0 | oclc=840278135 | ref=harv}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->
{{Functional analysis}}
| |||