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".

A closed operator that is used in practice is often densely defined.

Definition

be topological vector spaces.

A densely defined is a dense subset is a partial function whose domain.

.

Properties

The Hausdorff property ensures sequential convergence is unique. The metrizability property ensures that sequentially closed sets are closed. In functional analysis, these conditions typically hold, as most spaces under consideration are Fréchet space, or stronger than Fréchet. In particular, Banach spaces are Fréchet.

Examples

Sequence

be the Hilbert space of square-summable sequences. Define the diagonal operator

with domain

because the finitely supported sequences is closed and unbounded.

There exists a bounded inverse:

and, by the Neumann series argument, the resolvent set.

, the diagonal formula

.

is a densely defined, closed, unbounded operator with bounded inverse and nontrivial, unbounded spectrum.

Differentiation

of all real-valued, continuous functions denote the subspace consisting of all continuously differentiable functions with the supremum norm into a real Banach space. The differentiation operator.

is an example of an unbounded linear operator

Paley–Wiener

The Paley–Wiener integral is a standard 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 goes to the equivalence class This extension is the Paley–Wiener map.

See also

References