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- Closed_range_theorem abstract "In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires. Let and be Banach spaces, a closed linear operator whose domain is dense in , and the transpose of . The theorem asserts that the following conditions are equivalent: , the range of , is closed in , , the range of , is closed in , the dual of , , .Several corollaries are immediate from the theorem. For instance, a densely defined closed operator as above has if and only if the transpose has a continuous inverse. Similarly, if and only if has a continuous inverse.".
- Closed_range_theorem wikiPageID "23483982".
- Closed_range_theorem wikiPageRevisionID "590022928".
- Closed_range_theorem hasPhotoCollection Closed_range_theorem.
- Closed_range_theorem subject Category:Theorems_in_functional_analysis.
- Closed_range_theorem type Abstraction100002137.
- Closed_range_theorem type Communication100033020.
- Closed_range_theorem type Message106598915.
- Closed_range_theorem type Proposition106750804.
- Closed_range_theorem type Statement106722453.
- Closed_range_theorem type Theorem106752293.
- Closed_range_theorem type TheoremsInFunctionalAnalysis.
- Closed_range_theorem comment "In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires. Let and be Banach spaces, a closed linear operator whose domain is dense in , and the transpose of .".
- Closed_range_theorem label "Closed range theorem".
- Closed_range_theorem label "Satz vom abgeschlossenen Bild".
- Closed_range_theorem label "閉値域の定理".
- Closed_range_theorem label "闭值域定理".
- Closed_range_theorem sameAs Satz_vom_abgeschlossenen_Bild.
- Closed_range_theorem sameAs 閉値域の定理.
- Closed_range_theorem sameAs m.06w2yy6.
- Closed_range_theorem sameAs Q631561.
- Closed_range_theorem sameAs Q631561.
- Closed_range_theorem sameAs Closed_range_theorem.
- Closed_range_theorem wasDerivedFrom Closed_range_theorem?oldid=590022928.
- Closed_range_theorem isPrimaryTopicOf Closed_range_theorem.