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Difference between revisions of "Boundedly-compact set"

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''in a topological linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174401.png" />''
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''in a topological linear space $X$''
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174402.png" /> with the property that the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174403.png" /> of every bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174404.png" /> is compact and is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174405.png" /> (for a normed space in the strong (resp. weak) topology this is equivalent to the compactness (resp. weak compactness) of the intersections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017440/b0174406.png" /> 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.
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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 topology]] (resp. [[weak topology]]) this is equivalent to the compactness (resp. weak compactness) of the intersections of $M$ with balls). A [[Convex set|convex]] closed set in a normed space is boundedly compact if and only if it is [[Locally compact space|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 space]] which is boundedly compact (in itself) in the weak (resp. strong) topology is [[Reflexive space|reflexive]] (resp. a [[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 &amp; Winston  (1965)</TD></TR></table>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR>
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</table>
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[[Category:Topological spaces with richer structure]]
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{{TEX|done}}

Latest revision as of 17:07, 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 topology (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 space which is boundedly compact (in itself) in the weak (resp. strong) topology is reflexive (resp. a 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=13993
This article was adapted from an original article by L.P. Vlasov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article