Bounded tightness for weak topologies (Q1881506)

The principal result of the paper is Lemma 1 asserting that for two different Hausdorff locally convex topologies \(\delta\subset\tau\) on a vector space \(X\) there is a subset \(A\subset X\) such that \(0\in\overline{A}^\tau\) but \(0\notin\overline{B}^\tau\) for any \(\delta\)-bounded subset \(B\subset A\). This lemma is applied to studying the structure of locally convex spaces \(X\) whose weak topology is boundedly tight in the sense that for each set \(A\subset X\) and each point \(x\in\overline{A}^w\) there is a bounded set \(B\subset A\) with \(x\in\overline{B}^w\). It is shown that such a space \(X\) is (i) isomorphic to a subspace of the countable product \(\mathbb R^\omega\) of lines, provided that \(X\) is metrizable; (ii) isomorphic to \(\mathbb R^n\) for some \(n\leq \omega\), provided that \(X\) is complete; (iii) finite-dimensional, provided that \(X\) is a normed space.