Montel space


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

Characterizations

No infinite-dimensional Banach space is a Montel space, since these cannot satisfy the Heine-Borel property: the closed unit ball is closed and bounded, but not compact.
Montel spaces have the following properties:
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.
Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions. The space C∞ of smooth functions on an open set Ω in Rn is a Montel space equipped with the topology induced by the family of seminorms
for n = 1,2,… and K ranges over compact subsets of Ω, and α is a multi-index. Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions as K ranges over all compact subsets of Ω. The Schwartz space is also a Montel space.

Counter-examples

Every infinite-dimensional normed space is a barrelled space that is not a Montel space.
In particular, every infinite-dimensional reflexive 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.