Categorical algebra

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 linear operator T from one topological vector space, X, to another one, Y, is said to be densely defined if the domain of T is a dense subset of X.

Examples

• Consider the space C⁰([0, 1]; R) of all real-valued, continuous functions defined on the unit interval; let C¹([0, 1]; R) denote the subspace consisting of all continuously differentiable functions. Equip C⁰([0, 1]; R) with the supremum norm ||·||_∞; this makes C⁰([0, 1]; R) into a real Banach space. The differentiation operator D given by

$(\mathrm{D}u)(x)=u^{\prime }(x)$

is a densely defined operator from C⁰([0, 1]; R) to itself, defined on the dense subspace C¹([0, 1]; R). Note also that the operator D is an example of an unbounded linear operator, since

$u_n(x)=e^{-nx}$

has

$\frac{\Vert \mathrm{D}{u}_n{\Vert }_{\mathrm{\infty }}}{\Vert u_n{\Vert }_{\mathrm{\infty }}}=n.$

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C⁰([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 → 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 [f] 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.

References

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