# Definition:Bounded Mapping/Metric Space

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*This page is about mappings to metric spaces which are bounded. For other uses, see Definition:Bounded.*

## 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** if and only if $f \left({X}\right)$ is bounded in $M$.

## Also see

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.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples: Definition $2.2.14$