# Difference between revisions of "Approximate compactness"

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+ | A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a limit point $y\in M$. Approximate compactness of a given set ensures the existence of an element of best approximation for any $x\in X$. The concept of approximate compactness was introduced [[#References|[1]]] in connection with the study of Chebyshev sets (cf. [[Chebyshev set|Chebyshev set]]) in a Banach space, which made it possible to describe convex Chebyshev sets in certain spaces. In fact, let $X$ be a uniformly-convex smooth Banach space. For a Chebyshev set $M\subset X$ to be convex, it is necessary and sufficient that it be approximately compact. It follows, in particular, that the set of rational fractions with fixed degrees of the numerator and the denominator is not a Chebyshev set in the space $L_p$ ($1<p<\infty$) if the degree of the denominator is not smaller than one [[#References|[1]]]. | ||

For subsequent studies on this subject see [[#References|[2]]]. | For subsequent studies on this subject see [[#References|[2]]]. |

## Latest revision as of 11:04, 16 April 2014

A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a limit point $y\in M$. Approximate compactness of a given set ensures the existence of an element of best approximation for any $x\in X$. The concept of approximate compactness was introduced [1] in connection with the study of Chebyshev sets (cf. Chebyshev set) in a Banach space, which made it possible to describe convex Chebyshev sets in certain spaces. In fact, let $X$ be a uniformly-convex smooth Banach space. For a Chebyshev set $M\subset X$ to be convex, it is necessary and sufficient that it be approximately compact. It follows, in particular, that the set of rational fractions with fixed degrees of the numerator and the denominator is not a Chebyshev set in the space $L_p$ ($1<p<\infty$) if the degree of the denominator is not smaller than one [1].

For subsequent studies on this subject see [2].

#### References

[1] | N.V. Efimov, S.B. Stechkin, "Approximative compactness and Čebyšev sets" Soviet Math. Dokl. , 2 : 5 (1961) pp. 522–524 Dokl. Akad. Nauk SSSR , 140 : 3 (1961) pp. 522–524 |

[2] | A.L. Garkavi, "The theory of best approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian) |

**How to Cite This Entry:**

Approximate compactness.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Approximate_compactness&oldid=15388