Boundedly-compact set

in a topological linear space

A set with the property that the closure of every bounded subset is compact and is contained in (for a normed space in the strong (resp. weak) topology this is equivalent to the compactness (resp. weak compactness) of the intersections of 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)