View source for 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]]].

====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>