# Definition:Supremum Norm

## Definition

Let $S$ be a set.

Let $\left({X, \left\Vert{\cdot}\right\Vert}\right)$ be a normed vector space.

Let $\mathcal B$ be the set of bounded mappings $S \to X$.

For $f \in \mathcal B$ the supremum norm of $f$ on $S$ is:

$\left\Vert{f}\right\Vert_\infty = \sup \left\{ {\left\Vert{f \left({x}\right)}\right\Vert: x \in S}\right\}$

## Also known as

Other names include the sup norm, uniform norm or infinity norm.