On the Closed Graph Theorem and the Open Mapping Theorem - Ifsttar Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2019

On the Closed Graph Theorem and the Open Mapping Theorem

Résumé

Let E, F be two topological spaces and u : E → F be a map. If F is Haudorff and u is continuous, then its graph is closed. The Closed Graph Theorem establishes the converse when E and F are suitable objects of topological algebra, and more specifically topological groups, topological vector spaces (TVS’s) or locally vector spaces (LCS’s) of a special type. The Open Mapping Theorem, also called the Banach-Schauder theorem, states that under suitable conditions on E and F, if v : F → E is a continuous linear surjective map, it is open. When the Open Mapping Theorem holds true for v, so does the Closed Graph Theorem for u. The converse is also valid in most cases, but there are exceptions. This point is clarified. Some of the most important versions of the Closed Graph Theorem and of the Open Mapping Theorem are stated without proof but with the detailed reference.
Fichier principal
Vignette du fichier
R46.pdf (214.17 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02372404 , version 1 (20-11-2019)

Identifiants

  • HAL Id : hal-02372404 , version 1

Citer

Henri Bourlès. On the Closed Graph Theorem and the Open Mapping Theorem. 2019. ⟨hal-02372404⟩
162 Consultations
339 Téléchargements

Partager

Gmail Facebook X LinkedIn More