PlanetMath: bounded

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bounded

(Definition)

Let be a subset of $\mathbb{R}$ . We say that is bounded when there exists a real number such that $\vert x\vert<M$ for all $x\in X$ . When is an interval, we speak of a bounded interval.

This can be generalized first to $\mathbb{R}^n$ . We say that $X\subseteq \mathbb{R}^n$ is bounded if there is a real number such that $\Vert x\Vert<M$ for all $x\in X$ and $\Vert\cdot\Vert$ is the Euclidean distance between and .

This condition is equivalent to the statement: There is a real number such that $\Vert x-y\Vert<T$ for all $x,y\in X$ .

A further generalization to any metric space says that $X\subseteq V$ is bounded when there is a real number such that for all $x,y\in X$ , where is the metric on .

"bounded" is owned by yark. [ full author list (3) | owner history (3) ]

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See Also: Euclidean distance, metric space

Also defines:

bounded interval

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Cross-references: metric, metric space, Euclidean distance, interval, real number, subset
There are 42 references to this entry.

This is version 8 of bounded, born on 2003-10-15, modified 2006-05-20.
Object id is 4826, canonical name is BoundedInterval.
Accessed 6930 times total.

Classification:

AMS MSC:

54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

Pending Errata and Addenda

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