Supremum norm - Knowlpedia

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.

Remarks

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 $x=\pm 1$ ).

Comparison with an essential supremum

If $f$ is continuous on a space where every nonempty open set has positive measure, then $\sup|f|=\operatorname*{ess\,sup}|f|$ , including when both are infinite. Indeed, if $|f(x)|>a$ at a point, continuity gives a nonempty open set where $|f|>a$ , so an almost-everywhere bound by $a$ is impossible. Without continuity or this measure hypothesis, the two notions can differ.