Remark 3

Tag 033A, permanent. Cite it as [Bourbaki, Tag 033A].

Théories spectrales, chapter IV, § 4. Opérateurs partiels

Let $u$ be a symmetric partial operator on E. The operator $u$ is closable (prop. 7, b)). It satisfies dom( $u$ ) $\subset$ dom( $u^*$ ), and $u$ is self-adjoint if and only if dom( $u$ ) $=$ dom( $u^*$ ). Moreover, the closure $\overline{u}$ of $u$ is symmetric since $\overline{u}\subset u^*=\overline{u}^*$ (prop. 8).

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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Cites

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