View source for Approximate compactness

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]]].

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.L. Garkavi,   "The theory of best approximation in normed linear spaces"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 75–132  (In Russian)</TD></TR></table>