Definition:Bounded Metric Space

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

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