I'm stuck on this problem, namely I cannot find a bounded subset in $L^\infty(0,1)$ such that it is not mapped by the canonical inclusion $$j: L^\infty(0,1)\to L^1(0,1)$$ onto a relatively compact subset in $L^1(0,1)$. Can anybody provide me an example? Really I don't see the point.
My thoughts are wondering on the fact that the ball of $L^\infty(0,1)$ is norm dense in $L^1(0,1)$ so the inclusion cannot be compact, however, as i said, no practical examples come to my mind.
Thank you very much in advance.