Consider the space 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 ; this makes into a real Banach space. The differentiation operator given by
is a densely defined operator from to itself, defined on the dense subspace The operator is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator to the whole of
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 with adjoint there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from to ;\mathbb {R} ),}
under which goes to the equivalence class of in ;\mathbb {R} ).}
It can be shown that is dense in Since the above inclusion is continuous, there is a unique continuous linear extension ;\mathbb {R} )}
of the inclusion ;\mathbb {R} )}
to the whole of This extension is the Paley–Wiener map.