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?
Best Answer
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?