Definition 9

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

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

Let $q$ be a positive partial Hermitian form on E. We denote by $E_q$ the separated prehilbertian space dom( $q$ ) endowed with the scalar product

(x|y)_q=\langle x|y\rangle +q(x, y)

We denote by $\|x\|_q$ the norm of $x\in E_q$ . We say that the form $q$ is closed if $E_q$ is a Hilbert space.

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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