Approximate compactness
A property of a set 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) |
Approximate compactness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_compactness&oldid=31780