Corollary 1

Tag 08RI, permanent. Cite it as [Bourbaki, Tag 08RI].

Integration, IV, p. 28, § 3. LP spaces

If $\tilde{g}$ is the supremum of $\mathrm{H}$ in $L^p$ , then

\|\tilde{g}\|_p = \lim_{\tilde{f} \in \mathrm{H}} \|\tilde{f}\|_p = \sup_{\tilde{f} \in \mathrm{H}} \|\tilde{f}\|_p .

This follows from the continuity of the norm $\|\tilde{f}\|_p$ in $L^p$ , and the monotone limit theorem.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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