Шектелген және шектеулі жиынтықтардың ықшамдығы бар кеңістіктер

Barrelled space where closed and bounded subsets are compact
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.

A topological vector space (TVS) has the Heine–Borel property if every closed and bounded subset is compact. A Montel space is a barrelled topological vector space with the Heine–Borel property. Equivalently, it is an infrabarrelled semi Montel space where a Hausdorff locally convex topological vector space is called a semi Montel space or perfect if every bounded subset is relatively compact. A subset of a TVS is compact if and only if it is complete and totally bounded. A Fréchet–Montel space is a Fréchet space that is also a Montel space.

Montel spaces are paracompact and normal. Semi Montel spaces are quasi complete and semi reflexive while Montel spaces are reflexive. No infinite dimensional Banach space is a Montel space. This is because a Banach space cannot satisfy the Heine–Borel property: the closed unit ball is closed and bounded, but not compact. Fréchet Montel spaces are separable and have a bornological strong dual. A metrizable Montel space is separable. Fréchet–Montel spaces are distinguished spaces.

Every infinite dimensional normed space is a barrelled space that is not a Montel space. In particular, every infinite dimensional Banach space is not a Montel space. There exist Montel spaces that are not separable and there exist Montel spaces that are not complete. There exist Montel spaces having closed vector subspaces that are not Montel spaces.