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Montel space

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A barrelled space (in particular, a Fréchet space) in which each closed bounded set is compact. The space of all holomorphic functions in a domain , 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 , so is a Montel space. The space of all infinitely-differentiable functions in a domain , the space of all functions of compact support and the space 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 , and 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