# 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

${\displaystyle (\mathrm {D} u)(x)=u'(x)\,}$
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
${\displaystyle u_{n}(x)=e^{-nx}\,}$
has
${\displaystyle {\frac {\|\mathrm {D} u_{n}\|_{\infty }}{\|u_{n}\|_{\infty }}}=n.}$
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(EγR), under which j(f) ∈ j(E) ⊆ H goes to the equivalence class [f] of f in L2(EγR). 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(EγR) of the inclusion j(E) → L2(EγR) 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.