Definition:Supremum Norm

Definition
Let $S$ be a set, and $(X,\| \cdot \|)$ a normed vector space.

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

This is a vector space.

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


 * $\| f \|_{\infty} = \sup\{\|f(x)\| : x \in S \}$

This is indeed a norm by Supremum Norm is a Norm.

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