Michael selection theorem (original) (raw)
In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact space and Y a Banach space.Let be a lower hemicontinuous multivalued map with nonempty convex closed values.Then there exists a continuous selection of F.Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness.
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| dbo:abstract | In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact space and Y a Banach space.Let be a lower hemicontinuous multivalued map with nonempty convex closed values.Then there exists a continuous selection of F.Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness. (en) En mathématiques, le théorème de sélection de Michael, est un théorème d'analyse fonctionnelle démontré en 1956 par (en). Il s'énonce comme suit : Si X est un espace paracompact alors, toute multifonction hémicontinue inférieurement Γ, de X dans un espace de Banach E et à valeurs des convexes fermées non vides, possède une « sélection » continue, c'est-à-dire qu'il existe une application continue f : X → E telle que pour tout x de X, f(x) appartienne à Γ(x). Michael a aussi démontré la réciproque, si bien que cette propriété caractérise les espaces paracompacts (parmi les espaces séparés). (fr) |
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| rdfs:comment | In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact space and Y a Banach space.Let be a lower hemicontinuous multivalued map with nonempty convex closed values.Then there exists a continuous selection of F.Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness. (en) En mathématiques, le théorème de sélection de Michael, est un théorème d'analyse fonctionnelle démontré en 1956 par (en). Il s'énonce comme suit : Si X est un espace paracompact alors, toute multifonction hémicontinue inférieurement Γ, de X dans un espace de Banach E et à valeurs des convexes fermées non vides, possède une « sélection » continue, c'est-à-dire qu'il existe une application continue f : X → E telle que pour tout x de X, f(x) appartienne à Γ(x). Michael a aussi démontré la réciproque, si bien que cette propriété caractérise les espaces paracompacts (parmi les espaces séparés). (fr) |
| rdfs:label | Théorème de sélection de Michael (fr) Michael selection theorem (en) |
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