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 space
A partial function
is declared with the notation
For instance, the graph of a partial function
However, one exception to this is the definition of "closed graph".
A partial function
is said to have a closed graph if
in the product topology; importantly, note that the product space is
as it was defined above for ordinary functions.
is considered as an ordinary function (rather than as the partial function
), then "having a closed graph" would instead mean that
is a closed subset of
is a closed subset of
although 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 subspace
and a function (resp.
whose graph is equal to the closure of the set
is called a closure of
and necessarily extends
is a closable linear operator then a core or an essential domain of
of the graph of the restriction
is equal to the closure of the graph of
is equal to the closure of
Here are examples of closed operators that are not bounded.
The following properties are easily checked for a linear operator f : D(f) ⊆ X → Y between Banach spaces: