Montel space

E884920

locally convex space class topological vector space property

A Montel space is a type of locally convex topological vector space in which every closed and bounded set is compact, implying strong convergence and compactness properties useful in functional analysis and distribution theory.

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Statements (49)

Predicate	Object
instanceOf	locally convex space class ⓘ topological vector space property ⓘ
belongsToClass	barrelled locally convex spaces ⓘ semi-reflexive locally convex spaces ⓘ spaces where every closed bounded set is compact ⓘ
characterizedBy	every bounded sequence has a convergent subsequence in the strong topology (for Fréchet–Montel spaces) ⓘ every closed bounded set is compact in the given locally convex topology ⓘ
definedOn	locally convex topological vector space ⓘ
fieldOfStudy	distribution theory ⓘ functional analysis ⓘ topological vector spaces ⓘ
generalizes	Montel theorem for families of holomorphic functions NERFINISHED ⓘ
hasConsequence	bounded sets are relatively compact in every compatible locally convex topology ⓘ various versions of the Banach–Alaoglu theorem simplify ⓘ weak and strong boundedness coincide for many purposes ⓘ
hasDefiningProperty	every closed and bounded subset is compact ⓘ
hasExample	Fréchet space of holomorphic functions on a domain in ℂ^n ⓘ Schwartz space S(ℝ^n) NERFINISHED ⓘ certain spaces of distributions with appropriate topologies ⓘ space of real-analytic functions on an open set ⓘ space of test functions C_c^∞(Ω) ⓘ
hasHistoricalContext	introduced in connection with normal families of holomorphic functions ⓘ
hasTypicalTopology	locally convex topology defined by a family of seminorms ⓘ
impliesProperty	barrelled space ⓘ bornological space ⓘ every bounded set is precompact ⓘ every bounded subset is relatively compact ⓘ every closed bounded set is complete ⓘ every sequence in a bounded set has a convergent subsequence (for metrizable cases) ⓘ every weakly convergent sequence is strongly convergent (in many classical examples) ⓘ reflexive space (in the locally convex sense) ⓘ semi-reflexive space ⓘ
isStrongerThan	barrelled space ⓘ bornological space ⓘ reflexive locally convex space ⓘ
isWeakerThan	nuclear space (in many standard hierarchies) ⓘ
namedAfter	Paul Montel NERFINISHED ⓘ
oftenAssumedToBe	Hausdorff NERFINISHED ⓘ complete ⓘ
relatedConcept	DF-space (dual of a Fréchet space) ⓘ Fréchet–Montel space NERFINISHED ⓘ nuclear space ⓘ
topologicalDualProperty	the strong dual of a Montel space is Montel ⓘ the strong dual of a Montel space is barrelled ⓘ the strong dual of a Montel space is reflexive ⓘ
usedIn	compactness arguments in functional analysis ⓘ partial differential equations ⓘ theory of distributions ⓘ theory of holomorphic functions ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Produits tensoriels topologiques et espaces nucléaires → relatedConcept → Montel space ⓘ