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Difference between revisions of "Montel space"

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A [[Barrelled space|barrelled space]] (in particular, a [[Fréchet space|Fréchet space]]) in which each closed bounded set is compact. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648801.png" /> of all holomorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648802.png" />, with the topology of uniform convergence on compact sets, is a Fréchet space and, in view of a theorem of Montel (cf. [[Montel theorem|Montel theorem]], 2), every bounded sequence of holomorphic functions is relatively compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648803.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648804.png" /> is a Montel space. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648805.png" /> of all infinitely-differentiable functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648806.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648807.png" /> of all functions of compact support and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648808.png" /> of differentiable functions that are rapidly decreasing at infinity, are also Montel spaces in their natural topologies.
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A [[Barrelled space|barrelled space]] (in particular, a [[Fréchet space|Fréchet space]]) in which each closed bounded set is compact. The space $H(G)$ of all holomorphic functions in a domain $G$, with the topology of uniform convergence on compact sets, is a Fréchet space and, in view of a theorem of Montel (cf. [[Montel theorem|Montel theorem]], 2), every bounded sequence of holomorphic functions is relatively compact in $H(G)$, so $H(G)$ is a Montel space. The space $C^\infty(\Omega)=\mathcal E(\Omega)$ of all infinitely-differentiable functions in a domain $\Omega\subset\mathbf R^n$, the space $D(\Omega)$ of all functions of compact support and the space $S(\mathbf R^n)$ of differentiable functions that are rapidly decreasing at infinity, are also Montel spaces in their natural topologies.
  
A Montel space is reflexive (cf. [[Reflexive space|Reflexive space]]). The strong dual of a Montel space is a Montel space; in particular, the spaces of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m0648809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m06488010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064880/m06488011.png" /> are Montel spaces. A [[Normed space|normed space]] is a Montel space if and only if it is finite-dimensional.
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A Montel space is reflexive (cf. [[Reflexive space|Reflexive space]]). The strong dual of a Montel space is a Montel space; in particular, the spaces of generalized functions $\mathcal E'(\Omega)$, $D'(\Omega)$ and $S'(\Omega)$ are Montel spaces. A [[Normed space|normed space]] is a Montel space if and only if it is finite-dimensional.
  
 
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Latest revision as of 15:25, 17 July 2014

A barrelled space (in particular, a Fréchet space) in which each closed bounded set is compact. The space $H(G)$ of all holomorphic functions in a domain $G$, with the topology of uniform convergence on compact sets, is a Fréchet space and, in view of a theorem of Montel (cf. Montel theorem, 2), every bounded sequence of holomorphic functions is relatively compact in $H(G)$, so $H(G)$ is a Montel space. The space $C^\infty(\Omega)=\mathcal E(\Omega)$ of all infinitely-differentiable functions in a domain $\Omega\subset\mathbf R^n$, the space $D(\Omega)$ of all functions of compact support and the space $S(\mathbf R^n)$ of differentiable functions that are rapidly decreasing at infinity, are also Montel spaces in their natural topologies.

A Montel space is reflexive (cf. Reflexive space). The strong dual of a Montel space is a Montel space; in particular, the spaces of generalized functions $\mathcal E'(\Omega)$, $D'(\Omega)$ and $S'(\Omega)$ are Montel spaces. A normed space is a Montel space if and only if it is finite-dimensional.

References

[1] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[2] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)
[3] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)


Comments

References

[a1] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
[a3] H.H. Schaefer, "Topological vector spaces" , Springer (1971)
How to Cite This Entry:
Montel space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montel_space&oldid=32479
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article