Bourbaki

Remark 4

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

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

It is possible that αn(v∘u∘w)=0\alpha_n(v\circ u\circ w) = 0 even if αn(u)\alpha_n(u) is nonzero; in this case, αn(v∘u∘w)\alpha_n(v\circ u\circ w) is not a singular value of v∘u∘wv\circ u\circ w.

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.