Borel–Carathéodory theorem: Difference between revisions
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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 space|barrelled]] [[topological vector space]] where every [[closed set|closed]] and [[bounded set (topological vector space)|bounded set]] is [[compact space|compact]]. That is, it satisfies the [[Heine–Borel property]]. | |||
In classical [[complex analysis]], Montel's theorem asserts that the space of [[holomorphic function]]s on an [[open set|open]] [[connected space|connected]] subset of the [[complex number]]s has this property. | |||
== ' | Many Montel spaces of contemporary interest arise as spaces of [[test function]]s for a space of [[distribution (mathematics)|distributions]]. The space C<sup>∞</sup>(Ω) of [[smooth function]]s on an open set Ω in '''R'''<sup>''n''</sup> is a Montel space equipped with the topology induced by the family of [[seminorm]]s | ||
:<math>\|f\|_{K,n} = \sup_{|\alpha|\le n}\sup_{x\in K}\left|\partial^\alpha f(x)\right|</math> | |||
for ''n'' = 1,2,… and ''K'' ranges over compact subsets of Ω, and α is a [[multi-index]]. Similarly, is the space of [[compact support|compactly supported]] functions in an open set with the [[final topology]] of the family of inclusions <math>\scriptstyle{C^\infty_0(K)\subset C^\infty_0(\Omega)}</math> 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 space|nuclear]] quasi-complete [[barrelled space]] is Montel. | |||
* [[Frechet space|Frechet]] Montel spaces are separable and have a [[bornological space|bornological]] strong dual. | |||
==References== | |||
* {{springer|title=Montel space|id=p/m064880}} | |||
* {{cite book |last=Robertson |first=A.P. |coauthors= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | page=74}} | |||
* {{cite book | last = Schaefer | first = Helmuth H. | year = 1971 | title = Topological vector spaces | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6 | page=147 }} | |||
* {{cite book|first=François|last=Treves|title=Topological Vector Spaces, Distributions and Kernels|publisher=Dover|year=2006|isbn=978-0-486-45352-1}}. | |||
{{Functional Analysis}} | |||
[[Category:Topological vector spaces]] | |||
{{Mathanalysis-stub}} | |||
Revision as of 22:38, 21 November 2013
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
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 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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