Definition 4

Tag 0819, permanent. Cite it as [Bourbaki, Tag 0819].

Let $E$ be a hilbertian space and $u \in \mathscr{L}(E)$ . We say that $u$ is normal if it commutes with its adjoint $u^*$ .

For example, every automorphism $u$ of the hilbertian space $E$ is normal since we have $uu^* = u^*u = 1_E$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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