Densely Defined Operator - Examples

Examples

• Consider the space C0(R) of all real-valued, continuous functions defined on the unit interval; let C1(R) denote the subspace consisting of all continuously differentiable functions. Equip C0(R) with the supremum norm ||·||_∞; this makes C0(R) into a real Banach space. The differentiation operator D given by

is a densely defined operator from C0(R) to itself, defined on the dense subspace C1(R). Note also that the operator D is an example of an unbounded linear operator, since

has

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C0(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 → E with adjoint j = i∗ : E∗ → H, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from j(E∗) to L2(E, γ; R), under which j(f) ∈ j(E∗) ⊆ H goes to the equivalence class of f in L2(E, γ; R). It is not hard to show that j(E∗) is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H → L2(E, γ; R) of the inclusion j(E∗) → L2(E, γ; R) to the whole of H. This extension is the Paley–Wiener map.