Densely defined operator

Definition

Examples

• Consider the space C⁰ of all real-valued, continuous functions defined on the unit interval; let C¹ denote the subspace consisting of all continuously differentiable functions. Equip C⁰ with the supremum norm ||·||_∞; this makes C⁰ into a real Banach space. The differentiation operator D given by
• 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 from j to L², under which j ∈ j ⊆ H goes to the equivalence class of f in L². It is not hard to show that j is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H → L² of the inclusion j → L² to the whole of H. This extension is the Paley–Wiener map.