Bourbaki

Remark 1

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

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

Suppose that K=RK =\mathbf{R}. Let u∈L(E;F)u\in \mathscr{L}(E; F). One has u∈L1(E;F)u\in \mathscr{L}_1(E; F) if and only if u(C)∈L1(E(C);F(C))u_{(\mathbf{C})}\in \mathscr{L}_1(E_{(\mathbf{C})}; F_{(\mathbf{C})}); in this case, one has ∥u∥1=∥u(C)∥1\|u\|_1=\|u_{(\mathbf{C})}\|_1.

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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Machine translation of the English, by gpt-5-6-mini, gpt-5-mini, gpt-5.4, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.