Smooth Real Function times Derivative of Dirac Delta Distribution/Corollary

Theorem

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

Then in the distributional sense it holds that:

$x \delta' = - \delta$

Proof

From Smooth Real Function times Derivative of Dirac Delta Distribution:

$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$

where $\alpha$ is a smooth function.

If $\map \alpha x = x$, then:

$x \delta' = - \delta$

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions