Infunctional analysis and related areas ofmathematics, aMontel space, named afterPaul Montel, is anytopological vector space (TVS) in which an analog ofMontel's theorem holds. Specifically, a Montel space is abarrelled topological vector space in which everyclosed andbounded subset iscompact.

Atopological vector space (TVS) has theHeine–Borel property if everyclosed andbounded subset iscompact. AMontel space is abarrelled topological vector space with the Heine–Borel property. Equivalently, it is aninfrabarrelled semi-Montel space where aHausdorfflocally convex topological vector space is called asemi-Montel space orperfect if everybounded subset isrelatively compact.^{[note 1]} A subset of a TVS is compact if and only if it iscomplete andtotally bounded.AFréchet–Montel space is aFréchet space that is also a Montel space.

AseparableFréchet space is a Montel space if and only if eachweak-* convergent sequence in its continuous dual isstrongly convergent.^[1]

AFréchet 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 aFréchet space${\displaystyle X,}$ then${\displaystyle X}$ is Montel if and only if every sequence in${\displaystyle C^b(X)}$ that converges to zero in thecompact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of${\displaystyle X.}$^[2]

Semi-Montel spaces

A closed vector subspace of a semi-Montel space is again a semi-Montel space. The locally convexdirect sum of any family of semi-Montel spaces is again a semi-Montel space. Theinverse limit of an inverse system consisting of semi-Montel spaces is again a semi-Montel space. TheCartesian 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. Abarrelledquasi-completenuclear 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 strictinductive 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 evenreflexive.^[1] EveryFréchetSchwartz space is a Montel space.^[3]

Montel spaces areparacompact andnormal.^[4] Semi-Montel spaces arequasi-complete andsemi-reflexive while Montel spaces arereflexive.

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

Fréchet–Montel spaces aredistinguished spaces.

In classicalcomplex analysis, Montel's theorem asserts that the space ofholomorphic functions on anopenconnected subset of thecomplex numbers has this property.^{[citation needed]}

Many Montel spaces of contemporary interest arise as spaces oftest functions for a space ofdistributions. The space${\displaystyle C^{\mathrm{\infty }}(\mathrm{\Omega })}$ ofsmooth functions on an open set${\displaystyle \mathrm{\Omega }}$ in${\displaystyle {\mathbb{R}}^n}$ is a Montel space equipped with the topology induced by the family ofseminorms^[5]$${\displaystyle \Vert f{\Vert }_{K,n}=\underset{|\alpha |\le n}{sup}\underset{x\in K}{sup}\left|{\mathrm{\partial }}^{\alpha }f(x)\right|}$$for${\displaystyle n=1,2,\dots }$ and${\displaystyle K}$ ranges over compact subsets of${\displaystyle \mathrm{\Omega },}$ and${\displaystyle \alpha }$ is amulti-index. Similarly, the space ofcompactly supported functions in an open set with thefinal topology of the family of inclusions${\displaystyle C_0^{\mathrm{\infty }}(K)\subset C_0^{\mathrm{\infty }}(\mathrm{\Omega })}$ as${\displaystyle K}$ ranges over all compact subsets of${\displaystyle \mathrm{\Omega }.}$ TheSchwartz space is also a Montel space.

Every infinite-dimensionalnormed space is abarrelled space that isnot a Montel space.^[6] In particular, every infinite-dimensionalBanach space is not a Montel space.^[6] There exist Montel spaces that are notseparable and there exist Montel spaces that are notcomplete.^[6] There exist Montel spaces having closed vector subspaces that arenot Montel spaces.^[7]

• Barrelled space – Type of topological vector space
• Bornological space – Space where bounded operators are continuous
• Heine–Borel theorem – Subset of Euclidean space is compact if and only if it is closed and bounded
• LB-space
• LF-space – Topological vector space
• Nuclear space – Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces

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• Functional analysis
• Topological vector spaces

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