Supremum norm - Definition

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

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

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.

The occasion for the subscript "∞" is that

<math>\lim_{p\rightarrow\infty}\|f\|_p=\|f\|_\infty,<math>

where

<math>\|f\|_p=\left(\int_D \left|f\right|^p\right)^{1/p}<math>

where D is the domain of f.

The binary function

<math>d(f,g)=\|f-g\|_\infty<math>

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

<math>\lim_{n\rightarrow\infty}\|f_n-f\|_\infty=0.<math>

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