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.
Read more about this topic: Densely Defined Operator
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)