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".
[clarification needed] A closed operator that is used in practice is often densely defined.
A densely defined linear operator
from one topological vector space,
is a linear operator that is defined on a dense linear subspace
and takes values in
written
: dom (
{\displaystyle T:\operatorname {dom} (T)\subseteq X\to Y.}
Sometimes this is abbreviated as
when the context makes it clear that
might not be the set-theoretic domain of
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
into a real Banach space.
The differentiation operator
is a densely defined operator from
to itself, defined on the dense subspace
is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator
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
there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from
goes to the equivalence class
Since the above inclusion is continuous, there is a unique continuous linear extension
This extension is the Paley–Wiener map.