Proposition 7

Tag 081A, permanent. Cite it as [Bourbaki, Tag 081A].

For $u \in \mathscr{L}(E)$ to be normal, it is necessary and sufficient that $\|u(x)\| = \|u^*(x)\|$ for all $x \in E$ .

We define a hermitian form $\Phi$ on $E$ by

\Phi(x, y) = \langle uu^*(x)|y \rangle - \langle u^*u(x)|y \rangle .

For

u

to be normal, it is necessary and sufficient that

\Phi = 0

. By the polarization formulas (V, p. 2), this is equivalent to

\Phi(x, x) = 0

for all

x \in E

. The proposition now follows since

\Phi(x, x) = \|u^*(x)\|^2 - \|u(x)\|^2 .

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

Cited by

Nothing.