# Difference between revisions of "Approximately-compact set"

From Encyclopedia of Mathematics

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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 < | + | {{TEX|done}} |

+ | 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