Science Fair Project Encyclopedia

Science Fair Project Encyclopedia Contents Page

Uniform norm

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.

The occasion for the subscript "∞" is that

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

where

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

where D is the domain of f.

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

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

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

Science Fair Project Encyclopedia Contents Page

12-03-2008 10:22:39
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details

All Science Fair Projects

Science Fair Project Encyclopedia for Schools!

Science Fair Project Encyclopedia

Uniform norm