Editing Uniform norm (section)

=== 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>