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.^{[note 1]} 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.^[1]

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 ${\displaystyle X.}$  ^[2]

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.^[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.^[1] Every Fréchet Schwartz space is a Montel space.^[3]

Montel spaces are paracompact and normal.^[4] 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.^[1]

Fréchet–Montel spaces are distinguished spaces.

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.^{[citation needed]}

Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions. The space ${\displaystyle C^{\infty }(\Omega )}$  of smooth functions on an open set ${\displaystyle \Omega }$  in ${\displaystyle \mathbb {R} ^{n}}$  is a Montel space equipped with the topology induced by the family of seminorms^[5]

Counter-examples edit

Every infinite-dimensional normed space is a barrelled space that is not a Montel space.^[6] In particular, every infinite-dimensional Banach space is not a Montel space.^[6] There exist Montel spaces that are not separable and there exist Montel spaces that are not complete.^[6] There exist Montel spaces having closed vector subspaces that are not Montel spaces.^[7]

• Barrelled space – Type of topological vector space
• Bornological space – Space where bounded operators are continuous
• Heine–Borel theorem – Subset of Euclidean space is compact if and only if it is closed and bounded
• LB-space
• LF-space – Topological vector space
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces

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