Lemma 2

For every $k\in \mathscr{L}^{p'}(X\times Y)$, one has $N_{p', p'}(k) =\|k\|_{p'}$.

Let $k\in \mathscr{L}^{p'}(X\times Y)$. By INT, V, p. 96, § 8, No.$^o4$, Theorem 1, a), the set $Y'$ of elements $y\in Y$ such that the function $|k_y|^{p'}$ is not $\mu$-integrable is $\nu$-negligible; moreover, the function defined by $y\mapsto \mu(|k_y|^{p'})$ on Y - $Y'$ and zero on $Y'$ is $\nu$-integrable and satisfies

We denote by $\mathscr{L}^{p',q'}(X,Y, \mu, \nu )$, or simply by $\mathscr{L}^{p',q'}(X,Y)$, the complex vector space of $(\mu\otimes \nu$)-measurable functions $k$ of $X\times Y$ into $\mathbf{C}$ such that, for $\nu$-almost every $y\in Y$, the mapping $k_y$ belongs to $\mathscr{L}^{p'}$(X), and such that the mapping $k_{\circ}$ belongs to $\mathscr{L}_{L^{p'}(X)}^{q'}(Y, \nu )$ (INT, IV, p. 129, § 3, No.$^o4$, Definition 2). We endow $\mathscr{L}^{p',q'}(X,Y)$ with the seminorm $k\mapsto N_{p',q'}(k)$; note that one then has $N_{p',q'}(k) =\|k_{\circ}\|_{q'}$, where $k_{\circ}$ is regarded as a mapping with values in the Banach space $L^{p'}(X)$.