Remark

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

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

Let $\varphi \in \mathscr{S}(\mathbf{R}^n)$ . For every $k\in \mathbf{N}$ , one has

lim $_{\|x\|\rightarrow+\infty}\|x\|^k\varphi (x) = 0$ ,

since the function $x\mapsto \|x\|^{k+1}\varphi (x)$ is bounded.

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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