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 closed linear operator between Banach spaces is continuous; thus is a bounded operator. Hence, a closed linear operator that is used in practice is typically not defined on a Banach space or some other complete spaces but is often defined on a dense subspace.

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 which indicates that has prototype (that is, its domain is and its codomain is )

Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function is the set However, one exception to this is the definition of "closed graph". A partial function is said to have a closed graph if is a closed subset of in the product topology; importantly, note that the product space is and not as it was defined above for ordinary functions. In contrast, when 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 If is a closed subset of then it is also 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.

A linear operator is closable in if there exists a vector subspace containing and a function (resp. multifunction) whose graph is equal to the closure of the set in Such an is called a closure of in , is denoted by and necessarily extends

If is a closable linear operator then a core or an essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).

A bounded operator is a closed operator. Here are examples of closed operators that are not bounded.

• If is a Hausdorff TVS and is a vector topology on that is strictly finer than then the identity map a closed discontinuous linear operator.^[1]
• Consider the derivative operator where is the Banach space of all continuous functions on an interval If one takes its domain to be then is a closed operator, which is not bounded.^[2] On the other hand, if is the space of smooth functions scalar valued functions then will no longer be closed, but it will be closable, with the closure being its extension defined on

The following properties are easily checked for a linear operator f : D(f) ⊆ X → Y between Banach spaces:

• If A is closed then A − λId_D(f) is closed where λ is a scalar and Id_D(f) is the identity function;
• If f is closed, then its kernel (or nullspace) is a closed vector subspace of X;
• If f is closed and injective then its inverse f ⁻¹ is also closed;
• A linear operator f admits a closure if and only if for every x ∈ X and every pair of sequences x_• = (x_i)^∞
  _i=1 and y_• = (y_i)^∞
  _i=1 in D(f) both converging to x in X, such that both f(x_•) = (f(x_i))^∞
  _i=1 and f(y_•) = (f(y_i))^∞
  _i=1 converge in Y, one has lim_{i → ∞} fx_i = lim_{i → ∞} fy_i.

• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.