This note contains a short proof of the equivalence of the Schur and Dunford-Pettis properties in the class of discrete KB-spaces. We also present an operator characterization of the Schur property (Theorem 2) and we notice that Banach lattices which band hereditary coincide with Banach lattices having the Schur property. (This characterization is due to Popa (1977).) Moreover, the paper offers examples of Banach lattices with the positive Schur property and without the Schur property and which are not isomorphic to any AL-space.

1980 Mathematics Subject Classification (1985 revision): 46B30, 47B55, 47D15.