Definition:Multiplication of Distribution by Smooth Function

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Let $d \in \N$ be a natural number.

Let $\alpha \in \map {C^\infty} {\R^d}$ be a smooth function.

Let $T \in \map {\DD'} {\R^d}$ be a distribution.

Then the multiplication of a distribution by a smooth function $\alpha T$ is defined by:

$\alpha \map T \phi := \map T {\alpha \phi}$

where $\phi \in \map \DD {\R^d}$ is a test function.