Theorem 7

— Let $p$ and $q$ be two real numbers such that $1 \leq p < +\infty$, $1 \leq q < +\infty$. If a function $f$ belongs to $\mathscr{L}_F^p$ then the function $|f|^{(p/q)-1} \cdot f$ belongs to $\mathscr{L}_F^q$, and conversely.

By hypothesis, there exists a sequence $(f_n)$ of continuous functions with compact support such that $\sum_{n=1}^\infty N_p(f_n) < +\infty$ and $f(x) = \sum_{n=1}^\infty f_n(x)$ almost everywhere (No. 4, Th. 3). Set

the function $g_n$ is continuous with compact support; on the other hand,

where the numerical function $h \geqslant 0$ (finite or not) satisfies the inequality

by the countable convexity theorem. Moreover, $g_n(x)$ tends almost everywhere to $g(x) = |f(x)|^{(p/q)-1} \cdot f(x)$, therefore Lebesgue’s theorem shows that $g \in \mathscr{L}_F^q$. The converse is immediate, since $f = |g|^{(q/p)-1} \cdot g$.

It can be shown that the mapping $f \mapsto |f|^{\frac{p}{q}-1} \cdot f$ is a homeomorphism of $\mathscr{L}_F^p$ onto $\mathscr{L}_F^q$ (§ 6, Exer. 10).