Lemma 10

Tag 036V, permanent. Cite it as [Bourbaki, Tag 036V].

Théories spectrales, chapter IV, § 5. Opérateurs partiels normaux et théorème spectral

We have

\langle x|u(x)\rangle \geqslant \varrho_e\|x\|^2 \tag{18}

for every $x\in E_h\cap$ dom( $u$ ).

If $x\in E_h\cap$ dom( $u$ ), then the support of the spectral measure $\mu_x$ of $x$ relative to $u$ is contained in the interval $[\varrho_e,+\infty [$ (Prop. 9, a) of IV, p. 278), hence

\langle x|u(x)\rangle =\int_{\mathbf{R}}td\mu_x(t)\geqslant \varrho_e\|x\|^2

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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