# Definition:Supremum Norm

## Definition

Let $S$ be a set.

Let $\struct {X, \norm {\,\cdot\,} }$ 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 defined as:

$\norm f_\infty = \sup \set {\norm {\map f x}: x \in S}$

### Supremum Norm on Continuous on Closed Interval Real-Valued Functions

Let $I = \closedint a b$ be a closed real interval.

Let $\map \CC I$ be the space of real-valued functions, continuous on $I$.

Let $f \in \map \CC I$.

Let $\size {\, \cdot \,}$ denote the absolute value.

Suppose $\sup$ denotes the supremum of real-valued functions.

Then the supremum norm over $\map \CC I$ is defined as

$\displaystyle \norm {f}_\infty := \sup_{x \mathop \in I} \size {\map f x}$

## Also known as

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