# Densely defined operator

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".

## 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 dom(*T*) of *X* and takes values in *Y*, written *T* : dom(*T*) ⊆ *X* → *Y*. Sometimes this is abbreviated as *T* : *X* → *Y* when the context makes it clear that *X* might not be the set-theoretic domain of *T*.

## Examples

- Consider the space
*C*^{0}([0, 1];**R**) of all real-valued, continuous functions defined on the unit interval; let*C*^{1}([0, 1];**R**) denote the subspace consisting of all continuously differentiable functions. Equip*C*^{0}([0, 1];**R**) with the supremum norm ||·||_{∞}; this makes*C*^{0}([0, 1];**R**) into a real Banach space. The differentiation operator D given by

- is a densely defined operator from
*C*^{0}([0, 1];**R**) to itself, defined on the dense subspace*C*^{1}([0, 1];**R**). Note also that the operator D is an example of an unbounded linear operator, since

- has

- This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of
*C*^{0}([0, 1];**R**).

- 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*→*E*with adjoint*j*=*i*^{∗}:*E*^{∗}→*H*, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from*j*(*E*^{∗}) to*L*^{2}(*E*,*γ*;**R**), under which*j*(*f*) ∈*j*(*E*^{∗}) ⊆*H*goes to the equivalence class [*f*] of*f*in*L*^{2}(*E*,*γ*;**R**). It is not hard to show that*j*(*E*^{∗}) is dense in*H*. Since the above inclusion is continuous, there is a unique continuous linear extension*I*:*H*→*L*^{2}(*E*,*γ*;**R**) of the inclusion*j*(*E*^{∗}) →*L*^{2}(*E*,*γ*;**R**) to the whole of*H*. This extension is the Paley–Wiener map.

## References

- Renardy, Michael; Rogers, Robert C. (2004).
*An introduction to partial differential equations*. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0. MR 2028503.