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 closed operator that is used in practice is often densely defined.

Definition

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

Examples

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 D given by is a densely defined operator from to itself, defined on the dense subspace The operator \mathrm{D} is an example of an unbounded linear operator, since This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D 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 i : H \to E 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 [f] of f in It can be shown that is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension of the inclusion to the whole of H. This extension is the Paley–Wiener map.