Definition 2

Tag 030T, permanent. Cite it as [Bourbaki, Tag 030T].

Théories spectrales, chapter IV, § 3. Distributions et distributions tempérées

If $f\in \mathscr{D}'(U)$ is a distribution, $\partial^{\alpha}f$ is called the iterated partial derivative of order $\alpha$ of $f$ .

Thus, by definition,

\langle \partial^{\alpha}f, \varphi \rangle = (-1)^{|\alpha|}\langle f, \partial^{\alpha}\varphi \rangle

for every function $\varphi \in \mathscr{D}(U)$ . One has $\partial^{\alpha+\beta}=\partial^{\alpha}\circ \partial^{\beta}$ for all $\alpha$ and $\beta$ in $\mathbf{N}^n$ .

If $n= 1$ , the derivative of a distribution $f\in \mathscr{D}'(U)$ will also be denoted by $f'$ .

The definition is justified by the following lemma.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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