Definition:Bounded Metric Space

Definition
Let $M = \struct {A, d}$ be a metric space.

Let $M' = \struct {B, d_B}$ be a subspace of $M$.

Complex Plane
From Complex Plane is Metric Space, this concept can be applied directly to the complex plane:

Also defined as
Some sources place no emphasis on the fact that the subset $B$ of the underlying set $A$ of $M$ is in fact itself a subspace of $M'$, and merely refer to a bounded set.

This, however, glosses over the facts that:
 * $\text{(a)}$: from Subspace of Metric Space is Metric Space, any such subset is also a metric space by dint of the induced metric $d_B$
 * $\text{(b)}$: without reference to such a metric, boundedness is not defined.

Hence strives to ensure that boundedness is consistently defined in the context of a metric space, and not just a subset.

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

Also see

 * Equivalence of Definitions of Bounded Metric Space


 * Element in Bounded Metric Space has Bound


 * Definition:Totally Bounded Metric Space
 * Definition:Uniformly Bounded


 * Boundedness of Metric Space by Open Ball


 * Definition:Diameter of Subset of Metric Space


 * Definition:Bounded Mapping to Metric Space