Definitions

Uniform norm

In mathematical analysis, the uniform norm assigns to real- or complex-valued bounded functions f the nonnegative number

$|f|_infty=supleft\left\{,left|f\left(x\right)right|:xinmbox\left\{domain\right\} mbox\left\{of\right\} f,right\right\}.$

This norm is also called the supremum norm, the Chebyshev norm, and in the context of Lp spaces, the L norm.

If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question.

If f is a continuous function on a closed interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm.

In particular, for the case of a vector $x=\left(x_1,dots,x_n\right)$ in finite dimensional coordinate space, it takes the form

$|x|_infty=max\left\{ |x_1|, dots, |x_n| \right\}.$

The reason for the subscript "∞" is that

$lim_\left\{prightarrowinfty\right\}|f|_p=|f|_infty,$

where

$|f|_p=left\left(int_D left|fright|^p,dmuright\right)^\left\{1/p\right\}$

where D is the domain of f (and the integral amounts to a sum if D is a discrete set).

The binary function

$d\left(f,g\right)=|f-g|_infty$

is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence { fn : n = 1, 2, 3, ... } converges uniformly to a function f if and only if

$lim_\left\{nrightarrowinfty\right\}|f_n-f|_infty=0.,$

For complex continuous functions over a compact space, this turns it into a C* algebra.