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.

A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual is strongly convergent.

A Fréchet space ${\displaystyle X}$ is a Montel space if and only if every bounded continuous function ${\displaystyle X\to c_{0}}$ sends closed bounded absolutely convex subsets of ${\displaystyle X}$ to relatively compact subsets of ${\displaystyle c_{0}.}$ Moreover, if ${\displaystyle C^{b}(X)}$ denotes the vector space of all bounded continuous functions on a Fréchet space ${\displaystyle X,}$ then ${\displaystyle X}$ is Montel if and only if every sequence in ${\displaystyle C^{b}(X)}$ that converges to zero in the compact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of ${\displaystyle X.}$

Semi-Montel spaces

A closed vector subspace of a semi-Montel space is again a semi-Montel space. The locally convex direct sum of any family of semi-Montel spaces is again a semi-Montel space. The inverse limit of an inverse system consisting of semi-Montel spaces is again a semi-Montel space. The Cartesian product of any family of semi-Montel spaces (resp. Montel spaces) is again a semi-Montel space (resp. a Montel space).

Montel spaces

The strong dual of a Montel space is Montel. A barrelled quasi-complete nuclear space is a Montel space. Every product and locally convex direct sum of a family of Montel spaces is a Montel space. The strict inductive limit of a sequence of Montel spaces is a Montel space. In contrast, closed subspaces and separated quotients of Montel spaces are in general not even reflexive. Every Fréchet Schwartz space is a Montel space.