Distributional Derivative on Distributions is Linear Operator
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Theorem
The Distributional derivative on distributions is a linear operator.
Proof
Let $\phi, \psi \in \map \DD \R$ be test functions.
Let $\alpha \in \C$ be a complex number.
Let $T \in \map {\DD'} \R$ be a distribution.
Then:
\(\ds \map {T'} {\phi + \psi}\) | \(=\) | \(\ds - \map T {\paren {\phi + \psi}'}\) | Definition of Distributional Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds - \map T {\phi' + \psi'}\) | Sum Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds - \paren {\map T {\phi'} + \map T {\psi'} }\) | Definition of Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds - \map T {\phi'} - \map T {\psi'}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {T'} \phi + \map {T'} \psi\) | Definition of Distributional Derivative |
\(\ds \map {T'} {\alpha \phi}\) | \(=\) | \(\ds - \map T {\paren {\alpha \phi}'}\) | Definition of Distributional Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds - \map T {\alpha \phi'}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds - \alpha \map T {\phi'}\) | Definition of Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \alpha \map {T'} \phi\) | Definition of Distributional Derivative |
Thus the distributional derivative is a linear operator.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense