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

## Examples

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

${\displaystyle (\mathrm {D} u)(x)=u'(x)}$
is a densely defined operator from ${\displaystyle C^{0}([0,1];\mathbb {R} )}$ to itself, defined on the dense subspace ${\displaystyle C^{1}([0,1];\mathbb {R} ).}$ The operator ${\displaystyle \mathrm {D} }$ is an example of an unbounded linear operator, since
${\displaystyle u_{n}(x)=e^{-nx}\quad {\text{ has ))\quad {\frac {\left\|\mathrm {D} u_{n}\right\|_{\infty )){\left\|u_{n}\right\|_{\infty ))}=n.}$
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator ${\displaystyle D}$ to the whole of ${\displaystyle C^{0}([0,1];\mathbb {R} ).}$

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 ${\displaystyle i:H\to E}$ with adjoint ${\displaystyle j:=i^{*}:E^{*}\to H,}$ there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from ${\displaystyle j\left(E^{*}\right)}$ to ${\displaystyle L^{2}(E,\gamma ;\mathbb {R} ),}$ under which ${\displaystyle j(f)\in j\left(E^{*}\right)\subseteq H}$ goes to the equivalence class ${\displaystyle [f]}$ of ${\displaystyle f}$ in ${\displaystyle L^{2}(E,\gamma ;\mathbb {R} ).}$ It can be shown that ${\displaystyle j\left(E^{*}\right)}$ is dense in ${\displaystyle H.}$ Since the above inclusion is continuous, there is a unique continuous linear extension ${\displaystyle I:H\to L^{2}(E,\gamma ;\mathbb {R} )}$ of the inclusion ${\displaystyle j\left(E^{*}\right)\to L^{2}(E,\gamma ;\mathbb {R} )}$ to the whole of ${\displaystyle H.}$ This extension is the Paley–Wiener map.