Bourbaki

Definition 2

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

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

A locally convex space E is said to have the approximation property, or again to be an approximation space, if the identity mapping 1E1_E of E is adherent in the space Lpc(E)\mathscr{L}_{pc}(E) to the set Lf(E)\mathscr{L}^f(E) of continuous finite-rank endomorphisms of E.

In particular, every finite-dimensional locally convex space is an approximation space.

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.

Cited by

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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.