# 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) |

**How to Cite This Entry:**

Boundedly-compact set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Boundedly-compact_set&oldid=13993