Supremum norm

A norm on bounded functions given by the supremum of absolute values.

Supremum norm

The supremum norm of a bounded function $f:X\to\mathbb{R}$ is

\|f\|_\infty=\sup_{x\in X} |f(x)|.

Here $\sup$ denotes the supremum and $|\cdot|$ is the absolute value .

The supremum norm is the standard way to measure uniform size of functions and underlies the uniform metric . On many domains of interest (for example, a closed interval), continuous functions lie in the space of continuous functions and are bounded, so $\|\cdot\|_\infty$ is finite.

Examples:

For $f(x)=\sin x$ on $\mathbb{R}$ , $\|f\|_\infty=1$ .
For $f(x)=x^2$ on $[-1,1]$ , $\|f\|_\infty=1$ (the maximum is attained at $x=\pm 1$ ).