Definition:Bounded Metric Space

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

From the definition of a metric, $d: X \times X \to \R$ is a real-valued function.

Hence we can define that a metric space $\left({X, d}\right)$ is bounded if there exists $a \in X$ and $K \in \R$ such that $d \left({x, a}\right) \le K$ for all $x \in S$.

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

Metric Subspace
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 $M$) if $M'$ is bounded with respect to the subspace metric $d_Y$.

Mapping into Metric Space
Let $M$ be a metric space.

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

Then $f$ is called bounded if $f \left({X}\right)$ is bounded in $M$.

Mapping into Real Number Line
Note that as the real numbers form a 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.