I am thinking about the following problem. Let $1 \leq r < p \leq \infty$, and consider the spaces $L^r([0,1])$ and $L^p([0,1])$. A routine application of Holder shows that $L^p([0,1]) \subset L^r([0,1])$. My question is: Does there exist a bounded surjective linear map $T: L^p([0,1]) \rightarrow L^r([0,1])$? It is clear that in this case $L^p$ is a closed subspace of $L^r$. I tried using considerations using the open mapping theorem to show that this is not possible with no success. The case where $r=1$ can be resolved by considering the adjoint map $T^*: L^1([0,1])^* \rightarrow (L^p([0,1])^*$, which will be an injective map (since $T$ is surjective), which would imply $(L^1)^* = L^\infty$ embeds in $(L^p)^*$ which is impossible since $L^\infty$ is not seperable. However, this argument clearly does not extend to the general case.
More generally I’m interested in knowing (I’m not sure if there can be a clean answer to this): under what conditions does there exist a surjective bounded linear map from a closed subspace of a Banach space onto itself?