Function that is defined almost everywhere (mathematics)
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".
Examples
Consider the space
of all real-valued, continuous functions defined on the unit interval; let
denote the subspace consisting of all continuously differentiable functions. Equip
with the supremum norm
; this makes
into a real Banach space. The differentiation operator
given by
![{\displaystyle (\mathrm {D} u)(x)=u'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3cde9962fbb6a2db463674544ffc86f878481a)
is a densely defined operator from
to itself, defined on the dense subspace
The operator
is an example of an unbounded linear operator, since
![{\displaystyle u_{n}(x)=e^{-nx}\quad {\text{ has ))\quad {\frac {\left\|\mathrm {D} u_{n}\right\|_{\infty )){\left\|u_{n}\right\|_{\infty ))}=n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e97334acc99162fa3001951571d9086aeab0f58d)
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator
to the whole of
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
with adjoint
there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from
to
under which
goes to the equivalence class
of
in
It can be shown that
is dense in
Since the above inclusion is continuous, there is a unique continuous linear extension
of the inclusion
to the whole of
This extension is the Paley–Wiener map.