Uniform norm

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$\|x\|_\infty = 1$

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

$\|f\|_\infty=\sup\left\{\,\left|f(x)\right|:x\in\mbox{domain}\ \mbox{of}\ f\,\right\}.$

This norm is also called the supremum norm or the Chebyshev norm. If f is a continuous function on a closed interval, or more generally a compact set, then 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 $x = (x 1,..., x n)$ in finite dimensional coordinate space, it takes the form

$\|x\|_\infty=\max\{ |x_1|, ..., |x_n| \}.$

The reason for the subscript "∞" is that

$\lim_{p\rightarrow\infty}\|f\|_p=\|f\|_\infty,$

where

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

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

The binary function

$d(f,g)=\|f-g\|_\infty$

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

$\lim_{n\rightarrow\infty}\|f_n-f\|_\infty=0.$