A Banach space $H$ is said to have Schur's property if weak convergence of a sequence implies converge in norm. The most famous example of such a space is $\ell^1(\mathbb N)$, while $L^1[0,1]$ does not have this property.

My question is the following:

Is there a characterization of such spaces?

Is there a list of known examples, other than examples of the type $L^1(X)$?

Is it true that such space are not reflexive, when they are infinite dimensional?