Montel space

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.^{[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] $${\displaystyle \|f\|_{K,n}=\sup _{|\alpha |\leq n}\sup _{x\in K}\left|\partial ^{\alpha }f(x)\right|}$$ for ${\displaystyle n=1,2,\ldots }$ and ${\displaystyle K}$ ranges over compact subsets of ${\displaystyle \Omega ,}$ and ${\displaystyle \alpha }$ is a multi-index. Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions ${\displaystyle \scriptstyle {C_{0}^{\infty }(K)\subset C_{0}^{\infty }(\Omega )}}$ as ${\displaystyle K}$ ranges over all compact subsets of ${\displaystyle \Omega .}$ The Schwartz space is also a Montel space.

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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