Montel space

A topological vector space (TVS) has the Heine–Borel theorem#In the theory of topological vector spaces if every closed and bounded subset is compact.

A {{visible anchor|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 {{visible anchor|semi-Montel space}} or {{visible anchor|perfect}} if every bounded subset is relatively compact.A subset $S$ of a topological space $X$ is called relatively compact is its closure in $X$ is compact.

A subset of a TVS is compact if and only if it is complete and totally bounded.

A {{visible anchor|Fréchet–Montel space}} is a Fréchet space that is also a Montel space.

A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual is strongly convergent.{{sfn|Schaefer|Wolff|1999|pp=194-195}}

A Fréchet space $X$ is a Montel space if and only if every bounded continuous function $X\; \backslash to\; c\_0$ sends closed bounded absolutely convex subsets of $X$ to relatively compact subsets of $c\_0.$

Moreover, if $C^b(X)$ denotes the vector space of all bounded continuous functions on a Fréchet space $X,$ then $X$ is Montel if and only if every sequence in $C^b(X)$ that converges to zero in the compact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of $X.$

{{sfn|Lindström|1990|pp=191–196}}

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.{{sfn|Schaefer|Wolff|1999|pp=194-195}}

Every product and locally convex direct sum of a family of Montel spaces is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=194-195}}

The strict inductive limit of a sequence of Montel spaces is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=194-195}} In contrast, closed subspaces and separated quotients of Montel spaces are in general not even reflexive.{{sfn|Schaefer|Wolff|1999|pp=194-195}}

Every Fréchet Schwartz space is a Montel space.{{sfn|Khaleelulla|1982|pp=32-63}}

Montel spaces are paracompact and normal.{{cite web|title=Topological vector space|author=|date=|website=Encyclopedia of Mathematics|access-date=September 6, 2020|url=https://encyclopediaofmath.org/wiki/Topological_vector_space|quote=}}

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.{{sfn|Schaefer|Wolff|1999|pp=194-195}}

Fréchet–Montel spaces are distinguished spaces.

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.{{fact|date=March 2024}}

Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions.

The space $C^\{\backslash infty\}(\backslash Omega)$ of smooth functions on an open set $\backslash Omega$ in $\backslash R^n$ is a Montel space equipped with the topology induced by the family of seminorms{{harvnb|Hogbe-Nlend|Moscatelli|1981|p=235}}

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

for $n\; =\; 1,\; 2,\; \backslash ldots$ and $K$ ranges over compact subsets of $\backslash Omega,$ and $\backslash alpha$ is a multi-index. Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions $\backslash scriptstyle\{C^\backslash infty\_0(K)\backslash subset\; C^\backslash infty\_0(\backslash Omega)\}$ as $K$ ranges over all compact subsets of $\backslash Omega.$ The Schwartz space is also a Montel space.

Every infinite-dimensional normed space is a barrelled space that is {{em|not}} a Montel space.{{sfn|Khaleelulla|1982|pp=28-63}}

In particular, every infinite-dimensional Banach space is not a Montel space.{{sfn|Khaleelulla|1982|pp=28-63}}

There exist Montel spaces that are not separable and there exist Montel spaces that are not complete.{{sfn|Khaleelulla|1982|pp=28-63}}

There exist Montel spaces having closed vector subspaces that are {{em|not}} Montel spaces.{{sfn|Khaleelulla|1982|pp=103-110}}

• {{annotated link|Barrelled space}}
• {{annotated link|Bornological space}}
• {{annotated link|Heine–Borel theorem}}
• {{annotated link|LB-space}}
• {{annotated link|LF-space}}
• {{annotated link|Nuclear space}}

{{reflist|group=note}}

{{reflist|group=proof}}

{{reflist}}

• {{Edwards Functional Analysis Theory and Applications}}
• {{Hogbe-Nlend Bornologies and Functional Analysis}}
• {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} {{sfn whitelist|CITEREFHogbe-NlendMoscatelli1981}}
• {{Jarchow Locally Convex Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Köthe Topological Vector Spaces II}}
• {{cite journal|last=Lindström|first=Mikael|title=A note on Fréchet-Montel spaces|journal=Proceedings of the American Mathematical Society|publisher=American Mathematical Society (AMS)|volume=108|issue=1|date=1990-01-01|issn=0002-9939|doi=10.1090/s0002-9939-1990-0994780-8|pages=191–196|url=https://www.ams.org/journals/proc/1990-108-01/S0002-9939-1990-0994780-8/S0002-9939-1990-0994780-8.pdf}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}
• {{springer|title=Montel space|id=p/m064880}}

{{Functional analysis}}

{{Topological vector spaces}}

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Category:Functional analysis

Category:Topological vector spaces