$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? (Also I wonder whether the original proof of non-isomorphism was this one.)
Edit: to clear up any confusion, I am asking about the nonexistence of an isomorphism, not the nonexistence of an isometric isomorphism (which is basic, as pointed out in the comments).
Edit 2: the comment section seems to have degenerated into attempts to find alternate proofs of this fact (and debate over common misconceptions, such as whether a Banach space which is isomorphic to a dual space must itself be a dual space). That's great, although I still think the Schur property argument is the easiest one. However, this was not my question. Who first proved this?