# 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) ⊆ XY. Sometimes this is abbreviated as T : XY when the context makes it clear that X might not be the set-theoretic domain of T.

## Examples is a densely defined operator from C0([0, 1]; R) to itself, defined on the dense subspace C1([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 C0([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 L2(E, γ; R), under which j(f)  j(E)  H goes to the equivalence class [f] of f in L2(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  L2(E, γ; R) of the inclusion j(E)  L2(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.