Topologies on spaces of linear maps: Difference between revisions

Revision as of 17:53, 2 August 2020

Revision as of 17:54, 2 August 2020

Line 65:

==== Nets and uniform convergence ====

:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Let {{math|''f'' ∈ ''F''}} and let {{math|1=''f''_• = (''f''_''i'')_{''i'' ∈ ''I''}}} be a [[Net (mathematics)|net]] in {{mvar|F}}. Then for any subset {{mvar|G}} of {{mvar|T}}, say that {{math|''f''_•}} '''converges uniformly onto {{mvar|Gf}}''' toon {{mvar|fG}}''' if for every {{math|''N'' ∈ 𝒩}} there exists some {{math|''i''₀ ∈ ''I''}} such that for every {{math|''i'' ∈ ''I''}} satisfying {{math|''i'' ≥ ''i''₀}}, {{math|''f''_''i'' - ''f'' ∈ 𝒰(''G'', ''N'')}} (or equivalently, {{math|''f''_''i''(''g'') - ''f'' (''g'') ∈ ''N''}} for every {{math|''g'' ∈ ''G''}}).

{{Math theorem|name=Theorem{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=