Distributional Partial Derivatives Commute

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

Then in the distributional sense:


 * $\dfrac {\partial^2 T} {\partial x_i \partial x_j} = \dfrac {\partial^2 T} {\partial x_j \partial x_i}$

where


 * $i, j \in \N : 1 \le i, j \le d$

Proof
Let $\phi \in \map \DD {\R^d}$ be a test function.