Let $S$ be a nonempty set, and let $(V,\|\cdot\|_V)$ be a [normed vector space](/page/Normed%20Vector%20Space) over a scalar field $\mathbb K$, where $\mathbb K$ is either $\mathbb R$ or $\mathbb C$. Let $B(S,V)$ denote the set of all bounded functions $f:S\to V$, meaning that there exists $M\geq 0$ such that $\|f(s)\|_V\leq M$ for every $s\in S$. With pointwise addition and scalar multiplication, $B(S,V)$ is a [vector space](/page/Vector%20Space) over $\mathbb K$. Moreover, the function $\|\cdot\|_\infty:B(S,V)\to \mathbb R$ defined by

\begin{align*} \|f\|_\infty := \sup_{s\in S}\|f(s)\|_V \end{align*}

is a norm on $B(S,V)$.