Proposition 2

*If $f$ and $g$ are two mappings of $X$ into $F$, and $\alpha$ is any scalar $\neq 0$, then, for $1 \leq p < +\infty$,

(3)

(4)

For, the relation (3) follows at once from Def. 1 and the fact that $|\mu|^*$ is positively homogeneous; on the other hand, since $|f + g| \leq |f| + |g|$, the inequality (4) follows from Minkowski’s inequality (1) and the fact that $|\mu|^*$ is increasing.

We extend Def. 1 to the case of numerical functions, finite or not, defined on $X$, by again setting

for such a function $f$. One sees immediately that the relations (3) and (4) also hold for these functions when $f + g$ is defined on $X$ and $\alpha \neq 0$. Moreover: