Closed linear operator
In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property).It is a basic example of an unbounded operator.The closed graph theorem says a linear operator between Banach spaces is a closed operator if and only if it is a bounded operator.Hence, a closed linear operator that is used in practice is typically only defined on a dense subspace of a Banach space.It is common in functional analysis to consider partial functions, which are functions defined on a subset of some spaceA partial functionis declared with the notationFor instance, the graph of a partial functionHowever, one exception to this is the definition of "closed graph".A partial functionin the product topology; importantly, note that the product space isas it was defined above for ordinary functions.is considered as an ordinary function (rather than as the partial functionalthough the converse is not guaranteed in general.Definition: If X and Y are topological vector spaces (TVSs) then we call a linear map f : D(f) ⊆ X → Y a closed linear operator if its graph is closed in X × Y.if there exists a vector subspaceand a function (resp.whose graph is equal to the closure of the setis called a closure ofis a closable linear operator then a core or an essential domain ofof the graph of the restrictionis equal to the closure of the graph ofis equal to the closure ofHere are examples of closed operators that are not bounded.If one takes its domainis a closed operator, which is not bounded.of smooth functions scalar valued functions thenwill no longer be closed, but it will be closable, with the closure being its extension defined onThe following properties are easily checked for a linear operator f : D(f) ⊆ X → Y between Banach spaces: