Montel space

In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space where every closed and bounded set is compact. That is, it satisfies the Heine–Borel property.

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.

Many Montel spaces of contemporary interest arise as spaces of test functions for a space of distributions. The space C^∞(Ω) of smooth functions on an open set Ω in Rⁿ is a Montel space equipped with the topology induced by the family of seminorms

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

for n = 1,2,… and K ranges over compact subsets of Ω, and α is a multi-index. Similarly, the space of compactly supported functions in an open set with the final topology of the family of inclusions $\scriptstyle {C_{0}^{\infty }(K)\subset C_{0}^{\infty }(\Omega )}$ as K ranges over all compact subsets of Ω. The Schwartz space is also a Montel space.

No infinite-dimensional Banach space is a Montel space, since these cannot satisfy the Heine–Borel property: the closed unit ball is closed and bounded, but not compact.

They have the following properties:

The strong dual of a Montel space is Montel.
Montel spaces are reflexive.
A nuclear quasi-complete barrelled space is Montel.
Fréchet Montel spaces are separable and have a bornological strong dual.

Notes

References

Hazewinkel, Michiel, ed. (2001), "Montel space", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 74.
Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 147. ISBN 0-387-98726-6.
Treves, François (2006). Topological Vector Spaces, Distributions and Kernels. Dover. ISBN 978-0-486-45352-1. .

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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