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===Uniform structure===
====Nets and uniform convergence====▼
{{See also|Uniform space}}
For any <math>G \subseteq T</math> and <math>U \subseteq Y \times Y</math> be any [[Uniform space|entourage]] of <math>Y</math> (where <math>Y</math> is endowed with its [[Complete topological vector space#Canonical uniformity|canonical uniformity]]), let
<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}}
Let <math>f \in F</math> and let <math>f_{\bull} = \left(f_i\right)_{i \in I}</math> be a [[Net (mathematics)|net]] in <math>F.</math> Then for any subset <math>G</math> of <math>T,</math> say that <math>f_{\bull}</math> '''converges uniformly to <math>f</math> on <math>G</math>''' if for every <math>N \in \mathcal{N}</math> there exists some <math>i_0 \in I</math> such that for every <math>i \in I</math> satisfying <math>i \geq i_0,I</math> <math>f_i - f \in \mathcal{U}(G, N)</math> (or equivalently, <math>f_i(g) - f(g) \in N</math> for every <math>g \in G</math>).{{sfn|Jarchow|1981|pp=43-55}}
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