Topologies on spaces of linear maps: Difference between revisions

Revision as of 20:51, 27 November 2021

Revision as of 02:49, 14 December 2021

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=== Properties ===

;'''Hausdorffness'''

:'''Definition''':{{sfn|Schaefer|Wolff|1999|p=80}}A Ifsubset of a TVS <math>TX</math> whose [[linear span]] is a TVS[[dense thenset|dense wesubset]] say thatof <math>\mathcal{G}X</math> is '''totalsaid into <math>T</math>'''be if thea [[linearTotal spanset|total subset]] of <math>\bigcup_{G \in \mathcal{G}} GX.</math> is dense in <math>T.</math>

<li>If <math>\mathcal{G}</math> is anya collectionfamily of bounded subsets of a TVS <math>XT</math> whosethen union<math>\mathcal{G}</math> is said to be '''[[Total set|total in <math>XT</math>]]''' thenif everythe equicontinuous[[linear subsetspan]] of <math>L(X;\bigcup_{G Y)\in \mathcal{G}} G</math> is boundeddense in the <math>\mathcal{G}T.</math>-topology.{{sfn|Schaefer|Wolff|1999|p=8380}}</li>

If <math>F</math> is the vector subspace of <math>Y^T</math> consisting of all continuous linear maps that are bounded on every <math>G \in \mathcal{G},</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff if <math>Y</math> is Hausdorff and <math>\mathcal{G}</math> is total in <math>T.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}

;'''Completeness'''

For the following theorems, suppose that <math>X</math> is a topological vector space and <math>Y</math> is a [[locally convex]] Hausdorff spaces and <math>\mathcal{G}</math> is a collection of bounded subsets of <math>X</math> that covers <math>X,</math> is directed by subset inclusion, and satisfies the following condition: if <math>G \in \mathcal{G}</math> and <math>s</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that <math>s G \subseteq H.</math>

<ul>

<li><math>L_{\mathcal{G}}(X; Y)</math> is [[Complete topological vector space|complete]] if

{{ordered list|

|<math>X</math> is locally convex and Hausdorff,

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</ul>

;'''Boundedness'''

Let <math>X</math> and <math>Y</math> be topological vector spaces and <math>H</math> be a subset of <math>L(X; Y).</math>

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</ol>

If <math>\mathcal{G}</math> is a collection of bounded subsets of <math>X</math> whose union is [[Total set|total]] in <math>X</math> then every equicontinuous subset of <math>L(X; Y)</math> is bounded in the <math>\mathcal{G}</math>-topology.{{sfn|Schaefer|Wolff|1999|p=83}}

Furthermore,

Furthermore, if <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then

<ul>

<li>If <math>X</math> and <math>Y</math> are locally convex Hausdorff space and if <math>H</math> is bounded in <math>L_{\sigma}(X; Y)</math> (i.e.that is, pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>

<li>If <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces and if <math>X</math> is [[Quasi-complete space|quasi-complete]] (i.e.meaning that closed and bounded subsets are complete), then the bounded subsets of <math>L(X; Y)</math> are identical for all <math>\mathcal{G}</math>-topologies where <math>\mathcal{G}</math> is any family of bounded subsets of <math>X</math> covering <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>

<li></li>

<li>If <math>\mathcal{G}</math> is any collection of bounded subsets of <math>X</math> whose union is total in <math>X</math> then every equicontinuous subset of <math>L(X; Y)</math> is bounded in the <math>\mathcal{G}</math>-topology.{{sfn|Schaefer|Wolff|1999|p=83}}</li>

</ul>