Exercise 18, § 5

Let $A$ be a set contained in $L^r \cap L^s$ that is bounded in both $L^r$ and $L^s$ ($1 < r < s < +\infty$). Show that for every $p$ such that $r \leq p \leq s$, $A$ is bounded in $L^p$, and that the topologies induced on $A$ by the weakened topologies of the $L^p$ are identical (note that $\tilde{\mathscr{H}}(T)$ is dense in all of the spaces $L^q$ where $1 \leq q < +\infty$). Show that the property ceases to be valid if $r = 1$ (cf. Exer. 17 a)).