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