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 $$\operatorname{dom}(T)$$ of $$X$$ and takes values in $$Y,$$ written $$T : \operatorname{dom}(T) \subseteq X \to Y.$$ Sometimes this is abbreviated as $$T : X \to Y$$ 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 $$\|\,\cdot\,\|_\infty$$; this makes $$C^0([0, 1]; \R)$$ into a real Banach space. The differentiation operator $$D$$ given by $$(\mathrm{D} u)(x) = u'(x)$$ is a densely defined operator from $$C^0([0, 1]; \R)$$ to itself, defined on the dense subspace $$C^1([0, 1]; \R).$$ The operator $$\mathrm{D}$$ is an example of an unbounded linear operator, since $$u_n (x) = e^{- n x} \quad \text{ has } \quad \frac{\left\|\mathrm{D} u_n\right\|_{\infty}}{\left\|u_n\right\|_\infty} = n.$$ This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator $$D$$ 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 $$i : H \to E$$ with adjoint $$j := i^* : E^* \to H,$$ there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from $$j\left(E^*\right)$$ to $$L^2(E, \gamma; \R),$$ under which $$j(f) \in j\left(E^*\right) \subseteq H$$ goes to the equivalence class $$[f]$$ of $$f$$ in $$L^2(E, \gamma; \R).$$ It can be shown that $$j\left(E^*\right)$$ is dense in $$H.$$  Since the above inclusion is continuous, there is a unique continuous linear extension $$I : H \to L^2(E, \gamma; \R)$$ of the inclusion $$j\left(E^*\right) \to L^2(E, \gamma; \R)$$ to the whole of $$H.$$  This extension is the Paley–Wiener map.