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

Unbounded
Any space which is not bounded is described as unbounded.

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

Also see

 * Totally Bounded
 * Uniformly Bounded


 * Boundedness of Metric Space by Open Ball