Definition:Bounded Mapping/Metric Space

Definition
Let $M$ be a metric space.

Let $f: X \to M$ be a mapping from any set $X$ into $M$.

Then $f$ is a bounded mapping iff $f \left({X}\right)$ is bounded in $M$.

Also see

 * Bounded Real-Valued Function is Bounded in Metric Space

From Real Number Line is Metric Space, we can in theory consider defining boundedness on a real-valued function in terms of boundedness of a mapping into a metric space.

However, as a metric space is itself defined in terms of a real-valued function in the first place, this concept can be criticised as being a circular definition.