| dbo:abstract
|
- In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of that function is closed. The closed graph theorem has extensive application throughout functional analysis, because it can control whether a partially-defined linear operator admits continuous extensions. For this reason, it has been generalized to many circumstances beyond the elementary formulation above. (en)
|
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 26015 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:mathStatement
|
- Let be a linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a K-analytic space, and if the graph of is closed in then is continuous. (en)
- A linear operator from a barrelled space to a Fréchet space is continuous if and only if its graph is closed. (en)
- A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous. (en)
- If is an everywhere-defined linear operator between Banach spaces, then the following are equivalent:
# is continuous.
# is closed (en)
- Suppose that is a linear map whose graph is closed.
If is an inductive limit of Baire TVSs and is a webbed space then is continuous. (en)
- Let be linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a Souslin space, and if the graph of is a Borel set in then is continuous. (en)
- Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous. (en)
- If is a linear map between two F-spaces, then the following are equivalent:
# is continuous.
# has a closed graph.
# If in and if converges in to some then
# If in and if converges in to some then (en)
- Suppose that and are two topological vector spaces with the following property:
:If is any closed subspace of and is any continuous map of onto then is an open mapping.
Under this condition, if is a linear map whose graph is closed then is continuous. (en)
- A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous. (en)
- A linear map between two F-spaces is continuous if and only if its graph is closed. (en)
|
| dbp:name
|
- Theorem (en)
- Closed Graph Theorem (en)
- Borel Graph Theorem (en)
- Closed Graph Theorem for Banach spaces (en)
- Generalized Borel Graph Theorem (en)
|
| dbp:title
|
- Proof of closed graph theorem (en)
|
| dbp:urlname
|
- ProofOfClosedGraphTheorem (en)
|
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| rdfs:comment
|
- In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of that function is closed. (en)
|
| rdfs:label
|
- Closed graph theorem (functional analysis) (en)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
