Let $X$ be a non-empty set and $Y$ be a metric space. A function $f : X \to Y$ is bounded if $\im f$ is bounded.
We write
\[\fun_\text{bd}(X, Y) = \{f \in \fun(X, Y) : \text{$f$ is bounded}\}.\]
Let $X$ be a non-empty set and $Y$ be a metric space. A function $f : X \to Y$ is bounded if $\im f$ is bounded.
We write
\[\fun_\text{bd}(X, Y) = \{f \in \fun(X, Y) : \text{$f$ is bounded}\}.\]