Corollary 1

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

Integration, chapter IV, § 6. Convexity inequalities

— Let $F, G, H$ be three Banach spaces, and let $(u, v) \mapsto \Phi(u, v)$ be a continuous bilinear mapping of $F \times G$ into $H$ such that $|\Phi(u, v)| \leq |u| \cdot |v|$ . If $f \in \mathscr{L}_F^p$ and $g \in \mathscr{L}_G^q$ , then the function $\Phi(f, g)$ is integrable and

\left| \int \Phi(f, g)\, d\mu \right| \leq \int |\Phi(f, g)|\, d|\mu| \leq N_p(f) N_q(g).

For, $\Phi(f, g)$ is measurable (§ 5, No. 3, Cor. 5 of Th. 1); since $|\Phi(f, g)| \leq |f| \cdot |g|$ , the corollary follows from Th. 2 and the integrability criterion of § 5, No. 6, Th. 5.

Two special cases of Cor. 1 are important in applications:

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

Cited by

Corollary 5, § 6
Definition 1, § 5

Cites

§ 5, No. 3, Cor. 5 of Th. 1
§ 5, No. 6, Th. 5
Th. 2