  We prove that if $1 \leq p, q \leq \infty$, then the spaces $L_p +L_q$ and
$L_p \cap L_q$ are isomorphic if and only if $p = q$. In particular, $L_2
+L_{\infty}$ and $L_2 \cap L_{\infty}$ are not isomorphic which is an answer to
a question formulated in the paper S. V. Astashkin and L. Maligranda,
\textit{$L_p + L_{\infty}$ and $L_p \cap L_{\infty}$ are not isomorphic for all
$1 \leq p < \infty, p \neq 2$}, Proc. Amer. Math. Soc. 146 (2018), no. 5,
2181--2194.
