Corollary

— If $r, s, p$ are three numbers such that

then the intersection $\mathscr{L}_F^r \cap \mathscr{L}_F^s$ is contained in $\mathscr{L}_F^p$.

Note that in general the topologies induced on the intersection $\mathscr{L}_F^r \cap \mathscr{L}_F^s$ by the topologies of the $\mathscr{L}_F^p$ ($r < p < s$) are distinct. If no further hypothesis is made on $\mu$, the topologies induced on $\mathscr{L}_F^r \cap \mathscr{L}_F^s$ by those of $\mathscr{L}_F^r$ and $\mathscr{L}_F^s$ are in general not comparable (in other words, the ratio $N_r(f)/N_s(f)$ can take arbitrarily large values and arbitrarily small values in $\mathscr{L}_F^r \cap \mathscr{L}_F^s$; cf. Exer. 8).

Prop. 4 may be sharpened when $\mu$ is a bounded measure: