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.


$\blacksquare$


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