Corollary

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

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

For every $N\in L^2(X\times Y)$ , the linear mapping $u_N$ is a Hilbert–Schmidt mapping and one has Tr( $u^*_Nu_N$ ) $=\|N\|^2$ .

In fact, one has $\|u\|_2=\surd$ Tr( $u^*u$ ) for all $u\in \mathscr{L}_2(L^2(X); L^2(Y))$ .

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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