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".
Definition
A densely defined linear operator
from one topological vector space,
to another one,
is a linear operator that is defined on a dense linear subspace
of
and takes values in
written
Sometimes this is abbreviated as
when the context makes it clear that
might not be the set-theoretic domain of ![{\displaystyle T.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
is a densely defined operator from
to itself, defined on the dense subspace
The operator
is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator
to the whole of ![{\displaystyle C^{0}([0,1];\mathbb {R} ).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.
See also
References
- Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0. MR 2028503.