Corollary 2

— Let $F$ be a real (resp. complex) Banach space, $F'$ its strong dual (TVS, III, § 3, No. 1), and let $(z, z') \mapsto \langle z, z' \rangle$ be the canonical bilinear form on $F \times F'$. If $f \in \mathscr{L}_F^p$ and $g \in \mathscr{L}_{F'}^q$, then the real (resp. complex) function $\langle f, g \rangle$ is integrable and

For, $|\langle z, z' \rangle| \leq |z| \cdot |z'|$.

When $F$ is a real or complex Hilbert space, one knows that it can be canonically identified with its dual $F'$ (TVS, V, § 1, No. 7). Since the space $L_F^2$ is complete, we have the following result: