Difference between revisions of "Boundedly-compact set"
From Encyclopedia of Mathematics
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| − | ''in a topological linear space | + | ''in a topological linear space $X$'' |
| − | A set | + | A set $M$ with the property that the closure $\bar N$ of every bounded subset $N \subset M$ is compact and is contained in $M$ (for a normed space in the strong (resp. weak) topology this is equivalent to the compactness (resp. weak compactness) of the intersections of $M$ with balls). A convex closed set in a normed space is boundedly compact if and only if it is locally compact. Boundedly-compact sets have applications in the theory of approximation in Banach spaces; they have the property that an [[Element of best approximation|element of best approximation]] exists. A barrelled topological linear space which is boundedly compact (in itself) in the weak (resp. strong) topology is a reflexive (resp. Montel) space. A normed space which is boundedly compact is finite-dimensional. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.L. Klee, "Convex bodies and periodic homeomorphisms in Hilbert space" ''Trans. Amer. Math. Soc.'' , '''74''' (1953) pp. 10–43</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> V.L. Klee, "Convex bodies and periodic homeomorphisms in Hilbert space" ''Trans. Amer. Math. Soc.'' , '''74''' (1953) pp. 10–43</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 17:02, 17 October 2014
in a topological linear space $X$
A set $M$ with the property that the closure $\bar N$ of every bounded subset $N \subset M$ is compact and is contained in $M$ (for a normed space in the strong (resp. weak) topology this is equivalent to the compactness (resp. weak compactness) of the intersections of $M$ with balls). A convex closed set in a normed space is boundedly compact if and only if it is locally compact. Boundedly-compact sets have applications in the theory of approximation in Banach spaces; they have the property that an element of best approximation exists. A barrelled topological linear space which is boundedly compact (in itself) in the weak (resp. strong) topology is a reflexive (resp. Montel) space. A normed space which is boundedly compact is finite-dimensional.
References
| [1] | V.L. Klee, "Convex bodies and periodic homeomorphisms in Hilbert space" Trans. Amer. Math. Soc. , 74 (1953) pp. 10–43 |
| [2] | R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) |
How to Cite This Entry:
Boundedly-compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundedly-compact_set&oldid=33715
Boundedly-compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundedly-compact_set&oldid=33715
This article was adapted from an original article by L.P. Vlasov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article