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Difference between revisions of "Approximate compactness"

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A property of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128301.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128302.png" /> requiring that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128303.png" />, every minimizing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128304.png" /> (i.e. a sequence with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128305.png" />) has a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128306.png" />. Approximate compactness of a given set ensures the existence of an element of best approximation for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128307.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128308.png" /> be a uniformly-convex smooth Banach space. For a Chebyshev set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a0128309.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a01283010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012830/a01283011.png" />) if the degree of the denominator is not smaller than one [[#References|[1]]].
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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=31780
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article