Topologies on spaces of linear maps: Difference between revisions

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<math display=block>\mathcal{W}(G, U) ~:=~ \left\{(u, v) \in Y^T \times Y^T ~:~ (u(g), v(g)) \in U \; \text{ for every } g \in G\right\}.</math>

Given <math>G \subseteq T,</math> the family of all sets <math>\mathcal{W}(G, U)</math> as <math>U</math> ranges over any fundamental system of entourages of <math>Y</math> forms a fundamental system of entourages for a uniform structure on <math>Y^T</math> called {{em|the uniformity of uniform converges on <math>G</math>}} or simply {{em|the <math>G</math>-convergence uniform structure}}.{{sfn|Grothendieck|1973|pp=1-13}}

The {{em|the <math>\mathcal{G}</math>-convergence uniform structure}} is the least upper bound of all <math>G</math>-convergence uniform structures as <math>G \in \mathcal{G}</math> ranges over <math>\mathcal{G}.</math>{{sfn|Grothendieck|1973|pp=1-13}}

'''Nets and uniform convergence'''

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:<math>\mathcal{G}</math> is assumed to be closed with respect to the formation of subsets of finite unions of sets in <math>\mathcal{G}</math> (i.e. every subset of every finite union of sets in <math>\mathcal{G}</math> belongs to <math>\mathcal{G}</math>).

Some authors (e.g. Trèves {{sfn|Trèves|2006|loc=Chapter 32}}) require that <math>\mathcal{G}</math> be directed under subset inclusion and that it satisfy 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>

If <math>\mathcal{G}</math> is a [[bornology]] on <math>X,</math> which is often the case, then these axioms are satisfied.

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|whenever <math>u : X \to Y</math> is a linear map then <math>u</math> restricted to every set <math>G \in \mathcal{G}</math> is continuous implies that <math>u</math> is continuous,

}}</li>

<li>If <math>X</math> is a Mackey space then <math>L_{\mathcal{G}}(X; Y)</math> is complete if and only if both <math>X^{\prime}_{\mathcal{G}}</math> and <math>Y</math> are complete.</li>

<li>If <math>X</math> is [[Barrelled space|barrelled]] then <math>L_{\mathcal{G}}(X; Y)</math> is Hausdorff and [[quasi-complete]].</li>

<li>Let <math>X</math> and <math>Y</math> be TVSs with <math>Y</math> [[quasi-complete]] and assume that (1) <math>X</math> is [[Barreled space|barreled]], or else (2) <math>X</math> is a [[Baire space]] and <math>X</math> and <math>Y</math> are locally convex. If <math>\mathcal{G}</math> covers <math>X</math> then every closed [[Equicontinuous linear maps|equicontinuous subset]] of <math>L(X; Y)</math> is complete in <math>L_{\mathcal{G}}(X; Y)</math> and <math>L_{\mathcal{G}}(X; Y)</math> is quasi-complete.{{sfn|Schaefer|Wolff|1999|p=83}}</li>

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==𝒢-ℋ topologies on spaces of bilinear maps==

We will let <math>\mathcal{B}(X, Y; Z)</math> denote the space of separately continuous bilinear maps and <math>B(X, Y; Z)</math> denote the space of continuous bilinear maps, where <math>X, Y,</math> and <math>Z</math> are topological vector space over the same field (either the real or complex numbers).

In an analogous way to how we placed a topology on <math>L(X; Y)</math> we can place a topology on <math>\mathcal{B}(X, Y; Z)</math> and <math>B(X, Y; Z)</math>.

Let <math>\mathcal{G}</math> (respectively, <math>\mathcal{H}</math>) be a family of subsets of <math>X</math> (respectively, <math>Y</math>) containing at least one non-empty set.

Let <math>\mathcal{G} \times \mathcal{H}</math> denote the collection of all sets <math>G \times H</math> where <math>G \in \mathcal{G},</math> <math>H \in \mathcal{H}.</math>

We can place on <math>Z^{X \times Y}</math> the <math>\mathcal{G} \times \mathcal{H}</math>-topology, and consequently on any of its subsets, in particular on <math>B(X, Y; Z)</math> and on <math>\mathcal{B}(X, Y; Z)</math>.

This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math>-topology''' or as the '''topology of uniform convergence on the products <math>G \times H</math> of <math>\mathcal{G} \times \mathcal{H}</math>'''.

However, as before, this topology is not necessarily compatible with the vector space structure of <math>\mathcal{B}(X, Y; Z)</math> or of <math>B(X, Y; Z)</math> without the additional requirement that for all bilinear maps, <math>b</math> in this space (that is, in <math>\mathcal{B}(X, Y; Z)</math> or in <math>B(X, Y; Z)</math>) and for all <math>G \in \mathcal{G}</math> and <math>H \in \mathcal{H},</math> the set <math>b(G, H)</math> is bounded in <math>X.</math>

If both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consist of bounded sets then this requirement is automatically satisfied if we are topologizing <math>B(X, Y; Z)</math> but this may not be the case if we are trying to topologize <math>\mathcal{B}(X, Y; Z)</math>.

The <math>\mathcal{G}-\mathcal{H}</math>-topology on <math>\mathcal{B}(X, Y; Z)</math> will be compatible with the vector space structure of <math>\mathcal{B}(X, Y; Z)</math> if both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consists of bounded sets and any of the following conditions hold:

* <math>X</math> and <math>Y</math> are barrelled spaces and <math>Z</math> is locally convex.

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* {{annotated link|Uniform space}}

* {{annotated link|Weak topology}}

** {{annotated link|Vague topology}}

==References==

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* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{Functional Analysisanalysis}}

{{Duality and spaces of linear maps}}

{{DualityInLCTVSs}}

{{Topological vector spaces}}

{{TopologicalVectorSpaces}}

[[Category:Functional analysis]]