Topologies on spaces of linear maps: Difference between revisions

Revision as of 15:02, 30 June 2022

Revision as of 15:04, 30 June 2022

Line 101:

Throughout this section we will assume that <math>X</math> and <math>Y</math> are [[topological vector space]]s.

<math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by inclusion.

<math>L(X; Y)</math> will denote the vector space of all continuous linear maps from <math>X</math> into <math>Y.</math> If <math>L(X; Y)</math> is given the <math>\mathcal{G}</math>-topology inherited from <math>Y^X</math> then this space with this topology is denoted by <math>L_{\mathcal{G}}(X; Y)</math>.

'''Notation''':The <math>L(X;[[Dual Y)</math>space#Continuous willdual denotespace|continuous the vectordual space]] of alla continuoustopological linearvector maps fromspace <math>X</math> intoover the field <math>Y.\mathbb{F}</math> If(which <math>L(X;we will assume Yto be [[real numbers|real]] or [[complex numbers]])</math> is given the vector space <math>L(X; \mathcalmathbb{GF})</math>-topology inherited from <math>Y^X</math> then this space with this topologyand is denoted by <math>L_X^{\mathcal{Gprime}}(X; Y)</math>.

'''Notation''': The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space <math>X</math> over the field <math>\mathbb{F}</math> (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space <math>L(X; \mathbb{F})</math> and is denoted by <math>X^{\prime}</math>.

The <math>\mathcal{G}</math>-topology on <math>L(X; Y)</math> is compatible with the vector space structure of <math>L(X; Y)</math> if and only if for all <math>G \in \mathcal{G}</math> and all <math>f \in L(X; Y)</math> the set <math>f(G)</math> is bounded in <math>Y,</math> which we will assume to be the case for the rest of the article.

Line 114 ⟶ 112:

'''Assumptions that guarantee a vector topology'''

'''Assumption'''* (<math>\mathcal{G}</math> is directed): <math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by (subset) inclusion. That is, for any <math>G, H \in \mathcal{G},</math> there exists <math>K \in \mathcal{G}</math> such that <math>G \cup H \subseteq K</math>.

The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].

Line 120 ⟶ 118:

Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.

'''Assumption'''* (<math>N \in \mathcal{N}</math> are balanced): <math>\mathcal{N}</math> is a neighborhoods basis of the origin in <math>Y</math> that consists entirely of [[Balanced set|balanced]] sets.

The following assumption is very commonly made because it will guarantee that each set <math>\mathcal{U}(G, N)</math> is absorbing in <math>L(X; Y).</math>

'''Assumption'''* (<math>G \in \mathcal{G}</math> are bounded): <math>\mathcal{G}</math> is assumed to consist entirely of bounded subsets of <math>X.</math>

'''Other possible assumptions'''

The next theorem gives ways in which <math>\mathcal{G}</math> can be modified without changing the resulting <math>\mathcal{G}</math>-topology on <math>Y.</math>