In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".


A densely defined linear operator from one topological vector space, to another one, is a linear operator that is defined on a dense linear subspace of and takes values in written Sometimes this is abbreviated as when the context makes it clear that might not be the set-theoretic domain of


Consider the space of all real-valued, continuous functions defined on the unit interval; let denote the subspace consisting of all continuously differentiable functions. Equip with the supremum norm ; this makes into a real Banach space. The differentiation operator given by

is a densely defined operator from to itself, defined on the dense subspace The operator is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator to the whole of

The Paley–Wiener integral, on the other hand, is an example of a continuous extension of a densely defined operator. In any abstract Wiener space with adjoint there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from to under which goes to the equivalence class of in It can be shown that is dense in Since the above inclusion is continuous, there is a unique continuous linear extension of the inclusion to the whole of This extension is the Paley–Wiener map.

See also