# Definition:Bounded Metric Space

*This page is about Bounded in the context of Metric Space. For other uses, see Bounded.*

## Definition

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

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

### Definition 1

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

- $\exists a \in A, K \in \R: \forall x \in B: \map {d} {x, a} \le K$

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

### Definition 2

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

- $\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$

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

### Definition 3

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

- $\exists x \in A, \epsilon \in \R_{>0}: B \subseteq \map {B_\epsilon} x$

where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.

That is, $M'$ can be fitted inside an open ball.

### Definition 4

Let $a' \in A$.

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

- $\exists K \in \R: \forall x \in B: \map {d} {x, a'} \le K$

## 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:

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

## Euclidean Space

From Euclidean Space is Complete Metric Space, this concept can be applied directly to the Euclidean space:

Let $A \subseteq \R^n$ be a subset of a Euclidean space under the usual metric.

$A$ is **bounded (in $\R^n$)** if and only if :

- $\exists N \in \R: \forall x \in A: \size x \le N$

That is, every element of $A$ is within a finite distance of any point we may choose for the origin.

## Unbounded Metric Space

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

Let $M' = \struct {Y, d_Y}$ be a subspace of $M$.

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

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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**.

## Examples

## Also see

- Results about
**bounded metric spaces**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**bounded set** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**bounded set** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**bounded space**