BOUNDED TIGHTNESS FOR WEAK TOPOLOGIES

To the memory of our friend Klaus ABSTRACT. In every space for which there exists a strictly finer topology than its weak topology but with the same bounded sets (like for instance, all infinite dimensional Banach spaces, the space of distributions 0() or the space of analytic functions A() in an open set d, etc.) there is a set A such that 0 is in the weak closure of A but 0 is not in the weak closure of any bounded subset B of A. A consequence of this is that a Banach space X is finite dimensional if, and only if, the following property [P] holds: for each set A X and each x in the weak closure of A there is a bounded set B A such that x belongs to the weak closure of B. More generally, a complete locally convex space X satisfies property [P] if, and only if, either X is finite dimensional or linearly topologically isomorphic to . 1