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Hometotally bounded

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# totally bounded

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 the sumset $S+G$.

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 $\epsilon$.

# References

- 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

Keywords:

bounded,totally,totally bounded,total,bound,finite bound

Related:

MetricSpace, Bounded, Subset

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54E35*no label found*

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