Bourbaki

Remark 2

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

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

With the notations of the corollary, one has the formula

u(x)=∑i∈Iαi⟨ei∣x⟩fi(1) u(x) =\sum_{i\in I}\alpha_i\langle e_i|x\rangle f_i \tag{1}

for all x∈Ex\in E.

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.