Borel–Carathéodory theorem

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. That is, it satisfies the Heine–Borel property.

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^∞(Ω) of smooth functions on an open set Ω in Rⁿ is a Montel space equipped with the topology induced by the family of seminorms

$\|f\|_{K,n}=\sup _{|\alpha |\leq n}\sup _{x\in K}\left|\partial ^{\alpha }f(x)\right|$

for n = 1,2,… and K ranges over compact subsets of Ω, and α is a multi-index. Similarly, is the space of compactly supported functions in an open set with the final topology of the family of inclusions $\scriptstyle {C_{0}^{\infty }(K)\subset C_{0}^{\infty }(\Omega )}$ as K ranges over all compact subsets of Ω. The Schwartz space is also a Montel space.

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.

They have the following properties:

• The strong dual of a Montel spaces is Montel.
• Montel spaces are reflexive
• A nuclear quasi-complete barrelled space is Montel.
• Frechet Montel spaces are separable and have a bornological strong dual.

References

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