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 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)