Editing Uniform norm (section)

== Weaker structures inducing the topology of uniform convergence ==
=== Uniform metric ===
{{Main|Chebyshev distance}}
The '''uniform metric''' between two bounded functions <math>f,g\colon X\to Y</math> from a set <math>X</math> to a [[metric space]] <math>(Y,d_Y)</math> is defined by
:<math>d(f,g)=\sup_{x\in X}d_Y(f(x),g(x))</math>
The uniform metric is also called the '''{{visible anchor|Chebyshev metric}}''', after [[Pafnuty Chebyshev]], who was first to systematically study it. In this case, <math>f</math> is bounded precisely if <math>d(f,g)</math> is finite for some [[constant function]] <math>g</math>. If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called [[Metric (mathematics)#Generalized metrics|extended metric]] still allows one to define a topology on the function space in question; the convergence is then still the [[uniform convergence]]. In particular, a sequence <math>\left\{f_n : n = 1, 2, 3, \ldots\right\}</math> [[uniform convergence|converges uniformly]] to a function <math>f</math> if and only if
<math display=block>\lim_{n\rightarrow\infty}d(f_n,f)= 0.\,</math>

If <math>(Y,\|\|_Y)</math> is a [[normed space]], then it is a [[metric space]] in a natural way. The extended metric on <math>Y^X</math> induced by the uniform extended norm is the same as the uniform extended metric
:<math>d(f,g)=\sup_{x\in X}\|f(x)-g(x)\|_Y</math>
on <math>Y^X</math>

=== 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>.