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Montel space

## Summary

In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.

## Definition

A topological vector space (TVS) has the Heine–Borel property if every closed and bounded subset is compact. A Montel space is a barrelled topological vector space with the Heine–Borel property. Equivalently, it is an infrabarrelled semi-Montel space where a Hausdorff locally convex topological vector space is called a semi-Montel space or perfect if every bounded subset is relatively compact.[note 1] A subset of a TVS is compact if and only if it is complete and totally bounded. A Fréchet–Montel space is a Fréchet space that is also a Montel space.

## Characterizations

A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual is strongly convergent.[1]

A Fréchet space ${\displaystyle X}$  is a Montel space if and only if every bounded continuous function ${\displaystyle X\to c_{0}}$  sends closed bounded absolutely convex subsets of ${\displaystyle X}$  to relatively compact subsets of ${\displaystyle c_{0}.}$  Moreover, if ${\displaystyle C^{b}(X)}$  denotes the vector space of all bounded continuous functions on a Fréchet space ${\displaystyle X,}$  then ${\displaystyle X}$  is Montel if and only if every sequence in ${\displaystyle C^{b}(X)}$  that converges to zero in the compact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of ${\displaystyle X.}$  [2]

## Sufficient conditions

Semi-Montel spaces

A closed vector subspace of a semi-Montel space is again a semi-Montel space. The locally convex direct sum of any family of semi-Montel spaces is again a semi-Montel space. The inverse limit of an inverse system consisting of semi-Montel spaces is again a semi-Montel space. The Cartesian product of any family of semi-Montel spaces (resp. Montel spaces) is again a semi-Montel space (resp. a Montel space).

Montel spaces

The strong dual of a Montel space is Montel. A barrelled 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.[3]

## Properties

Montel spaces are paracompact and normal.[4] Semi-Montel spaces are quasi-complete and semi-reflexive while Montel spaces are reflexive.

No infinite-dimensional Banach space is a Montel space. This is because a Banach space cannot satisfy the Heine–Borel property: the closed unit ball is closed and bounded, but not compact. Fréchet Montel spaces are separable and have a bornological strong dual. A metrizable Montel space is separable.[1]

Fréchet–Montel spaces are distinguished spaces.

## 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.[citation needed]

Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions. The space ${\displaystyle C^{\infty }(\Omega )}$  of smooth functions on an open set ${\displaystyle \Omega }$  in ${\displaystyle \mathbb {R} ^{n}}$  is a Montel space equipped with the topology induced by the family of seminorms[5]

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

for ${\displaystyle n=1,2,\ldots }$  and ${\displaystyle K}$  ranges over compact subsets of ${\displaystyle \Omega ,}$  and ${\displaystyle \alpha }$  is a multi-index. Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions ${\displaystyle \scriptstyle {C_{0}^{\infty }(K)\subset C_{0}^{\infty }(\Omega )}}$  as ${\displaystyle K}$  ranges over all compact subsets of ${\displaystyle \Omega .}$  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.[6] In particular, every infinite-dimensional Banach space is not a Montel space.[6] There exist Montel spaces that are not separable and there exist Montel spaces that are not complete.[6] There exist Montel spaces having closed vector subspaces that are not Montel spaces.[7]

## Notes

1. ^ A subset ${\displaystyle S}$  of a topological space ${\displaystyle X}$  is called relatively compact is its closure in ${\displaystyle X}$  is compact.

## References

1. Schaefer & Wolff 1999, pp. 194–195.
2. ^ Lindström 1990, pp. 191–196.
3. ^ Khaleelulla 1982, pp. 32–63.
4. ^ "Topological vector space". Encyclopedia of Mathematics. Retrieved September 6, 2020.
5. ^ Hogbe-Nlend & Moscatelli 1981, p. 235
6. ^ a b c Khaleelulla 1982, pp. 28–63.
7. ^ Khaleelulla 1982, pp. 103–110.

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