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