Montel space

Summary

A Montel space is a barrelled space in which every closed bounded set is compact. This is a Heine–Borel property, which fails in every infinite-dimensional normed space. The name comes from Montel's theorem: the space of holomorphic functions on an open subset of $\C$ is a Montel space. Spaces of smooth functions and of test functions used in distribution theory are Montel spaces too, and Montel spaces are reflexive.