Definition:Bounded Metric Space/Euclidean
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Definition
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
Let $A \subseteq \R^n$ be a subset of a Euclidean space under the usual metric.
$A$ is unbounded (in $\R^n$) if and only if $A$ is not bounded (in $\R^n$).
Also see
- Results about bounded Euclidean 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