Editing Uniform norm (section)

=== Uniformity of uniform convergence ===
{{See also|Topologies on spaces of linear maps}}
Let <math>X</math> be a set and let <math>(Y,\mathcal E_Y)</math> be a [[uniform space]]. A sequence <math>(f_n)</math> of functions from <math>X</math> to <math>Y</math> is said to converge uniformly to a function <math>f</math> if for each entourage <math>E\in\mathcal E_Y</math> there is a natural number <math>n_0</math> such that, <math>(f_n(x),f(x))</math> belongs to <math>E</math> whenever <math>x\in X</math> and <math>n\ge n_0</math>. Similarly for a net. This is a convergence in a topology on <math>Y^X</math>. In fact, the sets
:<math>\{(f,g)\colon\forall x\in X\colon(f(x),g(x))\in E\}</math>
where <math>E</math> runs through entourages of <math>Y</math> form a fundamental system of entourages of a uniformity on <math>Y^X</math>, called the '''uniformity of uniform convergence''' on <math>Y^X</math>. The uniform convergence is precisely the convergence under its uniform topology.

If <math>(Y,d_Y)</math> is a [[metric space]], then it is by default equipped with the [[metric uniformity]]. The metric uniformity on <math>Y^X</math> with respect to the uniform extended metric is then the uniformity of uniform convergence on <math>Y^X</math>.