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

单词	TotallyBounded
释义	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
随便看	discrete density function DiscreteFourierTransform discrete Fourier transform DiscreteLogarithm discrete logarithm DiscreteSineTransform discrete sine transform DiscreteSpace discrete space DiscreteTimeFourierTransformInRelationWithContinuousTimeFourierTransform discrete time Fourier transform in relation with continuous time Fourier transform DiscreteValuation discrete valuation DiscreteValuationRing discrete valuation ring DiscreteWhiteNoise discrete white noise DiscretizationOfContinuousSystems discretization of continuous systems Discriminant discriminant discriminant Discriminant1 DiscriminantInAlgebraicNumberField discriminant in algebraic number field