View source for Boundedly-compact set

''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|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====
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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>
<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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[[Category:Topological spaces with richer structure]]

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