Bourbaki

Corollary

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

Théories spectrales, chapter III, § 1. Applications linéaires compactes

Let (xn)n∈N(x_n)_{n\in\mathbf{N}} be a sequence of points of E, which converges to a point xx of E for the weakened topology. The sequence (u(xn))n∈N(u(x_n))_{n\in\mathbf{N}} converges in F to u(x)u(x).

Indeed, the set consisting of the point xx and the points of the sequence (xn)n∈N(x_n)_{n\in\mathbf{N}} is a bounded subset of E (EVT, III, p. 3, cor. of prop. 2).

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.