Remark 3

Tag 02XI, permanent. Cite it as [Bourbaki, Tag 02XI].

Théories spectrales, chapter IV, § 1. Opérateurs compacts sur un espace hilbertien

Suppose that $K =\mathbf{C}$ . Let $u\in \mathscr{L}_1(E; E)$ be such that $u$ is positive. The norm $\|u\|_1$ of $u$ is the sum of the sequence $(\lambda_n(u))_{n\in I_E}$ (cor. 3 of IV, p. 165).

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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