every locally integrable function is a distribution
Suppose is an open set in and is a locally integrable function on , i.e., . Then the mapping
is a zeroth order distribution. (See parent entry for notation .)
(proof) (http://planetmath.org/T_fIsADistributionOfZerothOrder)
If and are both locally integrable functions on an open set ,
and , then it follows (see
this page (http://planetmath.org/TheoremForLocallyIntegrableFunctions)),
that almost everywhere. Thus, the mapping
is a linear injection when is equipped with
the usual equivalence relation
for an -space. For this reason,
one usually writes for the distribution .
Title | every locally integrable function is a distribution |
---|---|
Canonical name | EveryLocallyIntegrableFunctionIsADistribution |
Date of creation | 2013-03-22 13:44:25 |
Last modified on | 2013-03-22 13:44:25 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 8 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 46-00 |
Classification | msc 46F05 |