Proposition 2

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

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

The mapping $\widetilde{k}$ of $\mathscr{L}_{\mathbf{C}}^{\infty}(X, \mu)$ into $\mathscr{C}(Y)$ is compact.

For every $y$ and $y'$ in Y, one has

|\widetilde{k}(f)(y)|\leqslant \|k_y\|_1\|f\|_{\infty},|\widetilde{k}(f)(y)-\widetilde{k}(f)(y')|\leqslant \|k_y-k_{y'}\|_1\|f\|_{\infty}

By Ascoli's theorem (TG, X, p. 18, cor. 2), it follows that the image under $\widetilde{k}$ of the unit ball of $\mathscr{L}_{\mathbf{C}}^{\infty}(X;\mu)$ is relatively compact in $\mathscr{C}(Y)$ .

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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