Remark

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

Théories spectrales, chapter III, § 2. Exemples d’applications linéaires compactes

In the case $p=q= 2$ , it is generally more convenient to express the characterization of the integral operator $u_k$ by means of the scalar product: it is the unique mapping of $L^2(X)$ into $L^2(Y)$ such that

\langle g|u_k(f)\rangle =\int_{X\times Y}k(x, y)f(x)\overline{g(y)}d(\mu\otimes \nu )(x, y)

for all $f\in L^2(X)$ and $g\in L^2(Y)$ . Moreover, the adjoint of $u_k$ is the linear mapping $u_{k^*}$ , where $k^*\in L^2(Y\times X)$ satisfies $k^*(y, x) =k(x, y)$ for almost every $(y, x)\in Y\times X$ . In fact, for all $f\in L^2(X)$ and $g\in L^2$ (Y), one has

\langle g|u_k(f)\rangle =\int_{X\times Y}k(x, y)\overline{g(y)}f(x)d(\mu\otimes \nu )(x, y) =\langle u_{k^*}(g)|f\rangle

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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