$l^1$ 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?



Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?
This result was already published in the French (1932) edition of Banach's book "Theory of Linear Operations" (I do not know whether it was published in the Polish (1931) edition). On page 245 in the table you see two reasons why the spaces $\ell\_1$ and $L\_1(0,1)$ have different linear dimension:


1. Schur property for $\ell\_1$ and no Schur property for $L\_1$.
2. The fact that $\ell\_1$ is isomorphically embeddable into each of its infinite dimensional subspaces and the same is not true for $L\_1$.


Apparently for both statements about $L\_1$ Banach means that they follow from the Khinchin's inequality (1923), but I did not find the corresponding place in the book.


