Locally convex spaces and Schur type properties

In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and E _w have the same sequentially compact sets, where E _w is the space E with the weak topology, (3) E and E _w have the same compact sets, (4) E and E _w have the same countably compact sets, (5) E and E _w have the same pseudocompact sets, (6) E and E _w have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ ₁ and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in E _w .