Uniform norm

This article is about the function space norm. For the finite-dimensional vector space distance, see Chebyshev distance. For the uniformity norm in additive combinatorics, see Gowers norm.

The perimeter of the square is the set of points in ℝ² where the sup norm equals a fixed positive constant. For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.

In mathematical analysis, the uniform norm (or sup norm) assigns, to real- or complex-valued bounded functions ⁠$f$⁠ defined on a set ⁠$S$⁠, the non-negative number

$\|f\|_{\infty }=\|f\|_{\infty ,S}=\sup \left\{\,|f(s)|:s\in S\,\right\}.$

This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. The name "uniform norm" derives from the fact that a sequence of functions ⁠$\left\{f_{n}\right\}$⁠ converges to ⁠$f$⁠ under the metric derived from the uniform norm if and only if ⁠$f_{n}$⁠ converges to ⁠$f$⁠ uniformly.

If ⁠$f$⁠ is a continuous function on a closed and bounded 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, if ⁠$x$⁠ is some vector such that $x=\left(x_{1},x_{2},\ldots ,x_{n}\right)$ in finite dimensional coordinate space, it takes the form:

$\|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).$

This is called the $\ell ^{\infty }$-norm.