Definition:Bounded Metric Space

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


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:

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

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 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.

Also see

  • Results about bounded metric spaces can be found here.