So studying Qualifying Exam problems in Analysis I cam across this one:
For $1\lt r \lt p \lt s \lt \infty$ where $\mu$ denotes Lebesgue measure,
a) Construct a subspace of $L^p([0,1],\mu)$ such that $ \forall r$ the subspace is dense in $L^r([0,1],\mu)$ but not $L^p$.
b) Construct a subspace of $L^\infty([0,1,\mu)$ such that $ \forall s$ the subspace is dense in $L^p$ but not $L^s$.
Now $L^s\subset L^p \subset L^r$ since $\mu([0,1])\lt\infty$, so the issue is recognizing the norms are not equivalent for this to be possible. I'm just not terribly familiar with subspaces of $L^p$ for $p\neq2$. Is this just a matter of better knowing $L^p$ spaces or is there something bigger that I'm missing?