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