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totally bounded (Definition)

Let $ A$ be a subset of a topological vector space $ X$.

$ A$ is called totally bounded if , for each neighborhood $ G$ of 0, there exists a finite subset $ S$ of $ A$ with $ A$ contained in $ S + G$.

The definition can be restated in the following form when $ X$ is a metric space:

A set $ A \subseteq X$ is said to be totally bounded if for every $ \epsilon>0$, there exists a finite subset $ \{s_1,s_2,\ldots ,s_n\}$ of $ A$ such that $ A\subseteq \bigcup _{k=1} ^n B(s_k,\epsilon )$, where $ B(s_k,\epsilon)$ denotes the open ball around $ s_k$ with radius $ \epsilon$.

Bibliography

1
G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
2
A. Wilansky, Functional Analysis, Blaisdell Publishing Co., 1964
3
W. Rudin, Functional Analysis, 2nd ed. McGraw-Hill , 1973



"totally bounded" is owned by Mathprof. [ full author list (3) | owner history (4) ]
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See Also: metric space, bounded, subset

Keywords:  bounded, totally, totally bounded, total, bound, finite bound

Attachments:
totally bounded uniform space (Definition) by CWoo
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Cross-references: radius, open ball, metric space, contained, finite, neighborhood, topological vector space, subset
There are 5 references to this entry.

This is version 8 of totally bounded, born on 2002-11-19, modified 2006-09-04.
Object id is 3608, canonical name is TotallyBounded.
Accessed 6659 times total.

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