Are $L^p$ topologies compatible?

Question

Consider the subspace topologies on $L^p\cap L^q$ be induced by $L^p$ and $L^q$ respectively. Then, are these subspace topologies compatible?

Moreover, I'm curious about the special case $L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$.

Thank you in advance.

Answer 1

Let $f\in L^1\cap L^2$ and $f_t(x)=t^n\,f(t\,x)$. Then $f_t\in L^1\cap L^2$ and $$ \|f_t\|_1=\|f\|_1,\quad \|f_t\|_2=t^{n/2}\,\|f\|_2. $$ As $t\to0$, $f_t$ converges to $0$ in $L^2$, but not in $L^1$. The example can be changed to cover the case $L^p\cap L^q$.