Abstract.
In every Hausdorff locally convex 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 \( {\frak D}'(\Omega) \) or the space of analytic functions \( A(\Omega) \) in an open set , 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 \( A \subset X \) and each x in the weak closure of A there is a bounded set \( B \subset A \) 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 \( \mathbb{R}^\mathbb{N} \).
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Received: 11 June 2003
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Cascales, B., Raja, M. Bounded tightness for weak topologies. Arch. Math. 82, 324–334 (2004). https://doi.org/10.1007/s00013-003-0603-9
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DOI: https://doi.org/10.1007/s00013-003-0603-9