In mathematical analysis, the uniform norm (or sup norm) assigns, to real- or complex-valued bounded functions f {\displaystyle f} defined on a set S {\displaystyle S} , the non-negative number
‖ f ‖ ∞ = ‖ f ‖ ∞ , S = sup { | f ( s ) | : s ∈ S } . {\displaystyle \|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 { f n } {\displaystyle \left\{f_{n}\right\}} converges to f {\displaystyle f} under the metric derived from the uniform norm if and only if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly. If f {\displaystyle 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 {\displaystyle x} is some vector such that x = ( x 1 , x 2 , … , x n ) {\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)} in finite dimensional coordinate space, it takes the form:
‖ x ‖ ∞ := max ( | x 1 | , … , | x n | ) . {\displaystyle \|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}
This is called the ℓ ∞ {\displaystyle \ell ^{\infty }} -norm.
Definition Uniform norms are defined, in general, for bounded functions valued in a normed space. Let X {\displaystyle X} be a set and let ( Y , ‖ ‖ Y ) {\displaystyle (Y,\|\|_{Y})} be a normed space. On the set Y X {\displaystyle Y^{X}} of functions from X {\displaystyle X} to Y {\displaystyle Y} , there is an extended norm defined by
‖ f ‖ = sup x ∈ X ‖ f ( x ) ‖ Y ∈ [ 0 , ∞ ] . {\displaystyle \|f\|=\sup _{x\in X}\|f(x)\|_{Y}\in [0,\infty ].}
… excerpt ends here. Continue reading the full article.
