TITLE: Characterization of Schur's property
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: (Turning my comment into an answer here):
Regarding the third question: Yes, reflexive spaces with the Schur property need to be finite-dimensional. To see that, two ingredients suffice, once you notice that the Schur property can be phrased as "weakly sequentially compact sets are sequentially compact":

You need to show that in an infinite-dimensional space $X$, the unit ball is never (sequentially) compact. This follows from Riesz's lemma, which guarantees the existence of a sequence $(x_n) \in X^{\mathbb N}$ with $|x_n - x_m| \ge \frac 12$ for $n \ne m$.
You need to show that the unit ball in a reflexive space is weakly sequentially compact.

The proofs for both claims are elementary, unlike the Eberlein–Šmulian theorem; they can be found e.g. (in German, I'm afraid) in

Werner, Dirk. Funktionalanalysis. (German) [Functional analysis] Third, revised and extended edition. Springer-Verlag, Berlin, 2000. xii+501 pp. ISBN: 3-540-67645-7 MR1787146