Let $X$ be a non-empty set and $Y$ be a normed space. The following defines a norm on $\BMap(X,Y)$, called the supremum norm:
\[\lVert f\rVert_\sup = \sup\{\lVert f(x)\rVert : x\in X\}.\]
Let $X$ be a non-empty set and $Y$ be a normed space. The following defines a norm on $\BMap(X,Y)$, called the supremum norm:
\[\lVert f\rVert_\sup = \sup\{\lVert f(x)\rVert : x\in X\}.\]