TITLE: Does $L^1$ contain a subspace isomorphic to $c_0$?
QUESTION [5 upvotes]: Can any $L^1$ space, say $L^1(\mathbb{R})$, have some subspace isomorphic to $c_0$? I guess not but I don't see an argument right now.

REPLY [2 votes]: Here's another proof:
A non-reflexive subspace of $L_1$ contains an isomorphic copy of $\ell_1$. This follows from the Dunford-Pettis characterization of weakly compact subsets of $L_1$.  c.f., Kalton and Albiac, Topics in Banach Space Theory, Theorem 5.2.9. and Proposition 5.6.2; or Theorem 8 from  these notes by Joe Diestel.
On the other hand, $c_0$ does not contain $\ell_1$ isomorphically (otherwise, $c_0$ would have non-separable dual).