Corollary

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

Théories spectrales, chapter III, § 6. Propriétés spectrales des endomorphismes des espaces de Banach

Let E and F be Hilbert spaces and let $u$ be a continuous linear mapping from E into F. The linear mapping $u$ is compact if and only if the endomorphism $|u|$ of E is compact.

Let $(j,|u|)$ be the polar decomposition of $u$ (def. 4 of I, p. 140). Since

\surd

$u=j|u|$ and $|u|=u^*u$ (prop. 10 of I, p. 139), the equivalence results from prop. 3 of III, p. 5 and assertion b) of the preceding proposition.

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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