Let $E$ be a set, and let $B(E)$ denote the real [vector space](/page/Vector%20Space) of bounded functions $f: E \to \mathbb{R}$ with pointwise addition and scalar multiplication. Define $\|\cdot\|_\infty: B(E) \to [0,\infty)$ by
when $E$ is nonempty, and by $\|f\|_\infty = 0$ when $E$ is empty. Then $\|\cdot\|_\infty$ is a norm on $B(E)$. Equivalently, for every $f,g \in B(E)$ and every $\lambda \in \mathbb{R}$,