TITLE: Is the $L^1$-space dual to a Banach space
QUESTION [8 upvotes]: Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality
$$L^p=(L^{p*})^*,$$
where $1/p+1/p^*=1$.
Question. Is it known that the Banach space $L^1$ is not isomorphic to the dual space of any Banach space?
To avoid trivial cases let us assume that $L^1$ is infinite dimensional. I am particularly interested in the case when the total measure of $\Omega$ is finite.

REPLY [2 votes]: I would like to mention that for general measure spaces the answer is negative: The space $L_1(0,1)^{**}$ (by, e. g., Proposition  II.5.3 in Lindenstrauss--Tzafriri (1973)) is a dual $L_1$-space. 
Actually for purely atomic (even finite) measures the negative answer is obvious: consider a probability measure on a countable set.