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A subset {{mvar|H}} of {{mvar|F}} is [[Bounded set (topological vector space)|bounded]] in the {{math|𝒢}}-topology if and only if for every {{math|''G'' ∈ 𝒢}}, {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|Y}}.{{sfn | Schaefer | 1999 | p=81}}
===
;Pointwise convergence
If we let {{math|𝒢}} be the set of all finite subsets of {{mvar|T}} then the {{math|𝒢}}-topology on {{mvar|F}} is called the '''topology of pointwise convergence'''.
The topology of pointwise convergence on {{mvar|F}} is identical to the subspace topology that {{mvar|F}} inherits from {{math|''Y''<sup>''T''</sup>}} when {{math|''Y''<sup>''T''</sup>}} is endowed with the usual [[product topology]].
If {{mvar|X}} is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and {{math|C(''X'')}} is the space of all real (or complex) valued continuous functions on {{mvar|X}}, the the topology of pointwise convergence on {{math|C(''X'')}} is [[Metrizable TVS|metrizable]] if and only if {{mvar|X}} is countable.{{sfn | Jarchow | 1981 | pp=43-55}}
== 𝒢-topologies on spaces of continuous linear maps ==
Throughout this section we will assume that {{mvar|X}} and {{mvar|Y}} are [[topological vector space]]s.
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