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.
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.
[note 1] A subset of a TVS is compact if and only if it is complete and totally bounded.
A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual is strongly convergent.
is a Montel space if and only if every bounded continuous function
sends closed bounded absolutely convex subsets of
that converges to zero in the compact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of
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.
[1] Every product and locally convex direct sum of a family of Montel spaces is a Montel space.
[1] The strict inductive limit of a sequence of Montel spaces is a Montel space.
[1] In contrast, closed subspaces and separated quotients of Montel spaces are in general not even reflexive.
[3] Montel spaces are paracompact and normal.
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.
In classical complex analysis, Montel's theorem asserts that the space of holomorphic functions on an open connected subset of the complex numbers has this property.
of smooth functions on an open set
is a Montel space equipped with the topology induced by the family of seminorms[5]
Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions
[6] There exist Montel spaces that are not separable and there exist Montel spaces that are not complete.