# Definition:Bounded Metric Space

*This page is about metric spaces which are bounded. For other uses, see Definition:Bounded.*

## Contents

## 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.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**bounded set** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**bounded space**