Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both.
My idea:
Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger than $\Vert \cdot \Vert_1$, \begin{equation} B \text{ is compact in } (L^2,\Vert \cdot \Vert_2) \Rightarrow B \text{ is also compact in } (L^1,\Vert \cdot \Vert_1) \end{equation} therefore, $A \cap B$ is compact in $L^1$. However, I have no idea about how to determine if $A \cap B$ is compact in $L^2$ or not. Any hints or help? Thanks!