Let $X$ be a set, and let $K \in \{\mathbb R,\mathbb C\}$. Define $B(X;K)$ to be the set of all functions $f:X\to K$ for which there exists $M\ge 0$ such that $|f(x)|\le M$ for every $x\in X$. Equip $B(X;K)$ with pointwise addition and pointwise scalar multiplication over $K$. For $f\in B(X;K)$, define $\|f\|_\infty:=\sup_{x\in X}|f(x)|$, with the convention that this supremum is $0$ when $X=\varnothing$. Then $\|\cdot\|_\infty$ is a norm on $B(X;K)$, and $(B(X;K),\|\cdot\|_\infty)$ is complete. Hence $(B(X;K),\|\cdot\|_\infty)$ is a [Banach space](/page/Banach%20Space) over $K$.