Definition:Bounded Metric Space

Definition
Let $M = \left({X, d}\right)$ be a metric space.

Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$.

Then $M'$ is bounded (in $M$) iff there exists $a \in X$ and $K \in \R$ such that $d \left({x, a}\right) \le K$ for all $x \in Y$.

It follows immediately that, if $M'$ satisfies this condition for one $a \in X$, then it does so for all $a' \in X$, with $K$ replaced by $K^{\prime} = K + d \left({a, a^{\prime}}\right)$.

This is because $d \left({x, a}\right) \le K \implies d \left({x, a^{\prime}}\right) \le d \left({x, a}\right) + d \left({a, a^{\prime}}\right) \le K + d \left({a, a^{\prime}}\right)$.

Also known as
If the context is clear, it is acceptable to use the term bounded space for bounded metric space.

Also see

 * Definition:Totally Bounded
 * Definition:Uniformly Bounded


 * Boundedness of Metric Space by Open Ball


 * Definition:Diameter of Bounded Metric Subspace


 * Definition:Bounded Mapping to Metric Space