Remark

Tag 032T, permanent. Cite it as [Bourbaki, Tag 032T].

Théories spectrales, chapter IV, § 4. Opérateurs partiels

Let E and F be normed spaces over K and let $u$ be a partial operator from E into F.

1) The canonical injection of $E_u$ into E is continuous since one has $\|x\|\leqslant \|x\|_u$ for every $x\in E$ . In particular, every subspace of dom( $u$ ) which is closed in E is closed in $E_u$ .

2) If E and F are Hilbert spaces, then the space $E_u$ is a prehilbertian space, since the norm on $E_u$ arises from the positive Hermitian form

(x, y)\mapsto (x|y)_u=\langle x|y\rangle +\langle u(x)|u(y)\rangle

on dom( $u$ ).

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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