Proposition 7

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

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

Let $p\in [1,+\infty ]$ . The injection of $L^p(U)$ into $\mathscr{D}'(U)$ is continuous.

Let $f\in L^p(U)$ . Let K be a compact subset of U, and denote by $\varphi_K$ its characteristic function. For every test function $\varphi$ with support contained in K, Hölder's inequality implies

|\langle f, \varphi \rangle |=|\int_Uf \varphi d\mu|\leqslant N_p(f)N_q(\varphi )\leqslant N_p(f)N_q(\varphi_K)p_{0,K}(\varphi )

where $q$ is the conjugate exponent of $p$ .

In particular one can identify $\mathscr{D}(U)$ with a subspace of $\mathscr{D}'(U)$ .

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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