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Difference between revisions of "Approximately-compact set"

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A set having the property of [[Approximate compactness|approximate compactness]]. A metric projection on any approximately-compact Chebyshev set is continuous. Examples of approximately-compact sets include boundedly-compact sets and closed convex sets in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012880/a0128801.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012880/a0128802.png" />), and the set of rational fractions in which the degrees of the numerator and denominator are constant. In approximation theory and in the theory of ill-posed problems frequent use is made of spaces in which all closed sets are approximately compact.
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A set having the property of [[Approximate compactness|approximate compactness]]. A metric projection on any approximately-compact Chebyshev set is continuous. Examples of approximately-compact sets include boundedly-compact sets and closed convex sets in the spaces $L_p$ ($0<p<\infty$), and the set of rational fractions in which the degrees of the numerator and denominator are constant. In approximation theory and in the theory of ill-posed problems frequent use is made of spaces in which all closed sets are approximately compact.
  
 
====References====
 
====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. 1226–1228  ''Dokl. Akad. Nauk SSSR'' , '''140''' :  3  (1961)  pp. 522–524</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Vlasov,  "Approximative properties of sets in normed linear spaces"  ''Russian Math. Surveys'' , '''28''' :  6  (1973)  pp. 1–66  ''Uspekhi Mat. Nauk'' , '''28''' :  6  (1973)  pp. 3–66</TD></TR></table>
 
<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. 1226–1228  ''Dokl. Akad. Nauk SSSR'' , '''140''' :  3  (1961)  pp. 522–524</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Vlasov,  "Approximative properties of sets in normed linear spaces"  ''Russian Math. Surveys'' , '''28''' :  6  (1973)  pp. 1–66  ''Uspekhi Mat. Nauk'' , '''28''' :  6  (1973)  pp. 3–66</TD></TR></table>

Latest revision as of 11:05, 16 April 2014

A set having the property of approximate compactness. A metric projection on any approximately-compact Chebyshev set is continuous. Examples of approximately-compact sets include boundedly-compact sets and closed convex sets in the spaces $L_p$ ($0<p<\infty$), and the set of rational fractions in which the degrees of the numerator and denominator are constant. In approximation theory and in the theory of ill-posed problems frequent use is made of spaces in which all closed sets are approximately compact.

References

[1] N.V. Efimov, S.B. Stechkin, "Approximative compactness and Čebyšev sets" Soviet Math. Dokl. , 2 : 5 (1961) pp. 1226–1228 Dokl. Akad. Nauk SSSR , 140 : 3 (1961) pp. 522–524
[2] L.P. Vlasov, "Approximative properties of sets in normed linear spaces" Russian Math. Surveys , 28 : 6 (1973) pp. 1–66 Uspekhi Mat. Nauk , 28 : 6 (1973) pp. 3–66
How to Cite This Entry:
Approximately-compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximately-compact_set&oldid=15348
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article