Let $X$ be a non-empty set and $Y$ be a metric space. A mapping $f:X\to Y$ is bounded if $\ran f$ is bounded.
We write
\[\BMap(X,Y) = \{f\in\Map(X,Y) : \text{$f$ is bounded}\}.\]
Let $X$ be a non-empty set and $Y$ be a metric space. A mapping $f:X\to Y$ is bounded if $\ran f$ is bounded.
We write
\[\BMap(X,Y) = \{f\in\Map(X,Y) : \text{$f$ is bounded}\}.\]