Topologies on spaces of linear maps: Difference between revisions

Revision as of 20:26, 3 September 2022

Revision as of 16:09, 25 December 2022

Line 69:

<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'''