Bourbaki

Remark 1

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

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

Since the sequence (αn(u))n∈IE(\alpha_n(u))_{n\in I_E} is decreasing, the set J is either equal to IEI_E, or equal to a segment {0,...,m}\{0, . . . , m\} in IEI_E where m∈Nm\in \mathbf{N}. This latter case holds if and only if uu is of finite rank.

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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Cites

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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.