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 (that is, it satisfies the Heine-Borel property).

Characterizations
  • A separable Fréchet space is a Montel space if and only if each weakly convergent subsequence in is strongly convergent.[1]

Sufficient conditions
  • The strong dual of a Montel space is Montel.
  • A nuclear quasi-complete barrelled space is Montel.
  • A barreled 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.[2]

Properties

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:

Examples

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

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.[3] In particular, every infinite-dimensional reflexive Banach space is not a Montel space.[3] There exist Montel spaces that are not separable and there exist Montel spaces that are not complete.[3] There exist Montel spaces having closed vector subspaces that are not Montel spaces.[4]

Notes
  1. Schaefer 1999, pp. 194-195.
  2. Khaleelulla 1982, pp. 32-63.
  3. Khaleelulla 1982, pp. 28-63.
  4. Khaleelulla 1982, pp. 103-110.

References
    • Hazewinkel, Michiel, ed. (2001) [1994], "Montel space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
    • Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
    • Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 74.
    • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
    • Trèves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)
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