# Definition:Bounded Metric Space

## Definition

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

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

### Definition 1

$M'$ is bounded (in $M$) if and only if:

$\exists a \in A, K \in \R: \forall x \in B: d \left({x, a}\right) \le K$

That is, there exists an element of $A$ within a finite distance of all elements of $B$.

### Definition 2

$M'$ is bounded if and only if:

$\exists K \in \R: \forall x, y \in M': d \left({x, y}\right) \le K$

That is, there exists a finite distance such that all pairs of elements of $B$ are within that distance.

## Complex Plane

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

Let $D$ be a subset of the complex plane $\C$.

Then $D$ is bounded (in $\C$) if and only if there exists $M \in \R$ such that:

$\forall z \in D: \cmod z \le M$

## Unbounded Metric Space

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 unbounded (in $M$) if and only if $M'$ is not bounded in $M$.

## Also known as

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

## Also see

• Results about bounded metric spaces can be found here.