Remark 2

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

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

Let $u$ be a nuclear mapping of E into F. Since the mapping $u$ is of Hilbert–Schmidt type, it is compact (cor. 2 of IV, p. 165). Moreover, Proposition 14 implies that for every $v\in \mathscr{L}_2(E; F)$ , one has

|\langle v|u\rangle |\leqslant \|v\| \|u\|_1 \tag{11}

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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