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 $\ell^1(X)$?


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



Characterization of Schur's property
Rosenthal's $\ell\_1$ theorem (Google) says that every bounded sequence in a Banach space contains a subsequence that is either weakly Cauchy or is equivalent to the unit vector basis of $\ell\_1$. From this you get that a Banach space has the Schur property iff for every $\epsilon > 0$, every $\epsilon$ separated bounded sequence has a subsequence that is equivalent to the unit vector basis of $\ell\_1$. That answers your first question. 


The answer to the third question is, obviously, yes. Unit balls of reflexive spaces are weakly sequentially compact by the Eberlein-Smulian theorem. 


As for the second question, there are many examples, but because of the characterization above, all are in some sense constructed from $\ell\_1$. What are you looking for? 


