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A remark on a problem of Klee. (English) Zbl 0895.46002

Let \(X\) be a topological vector space. \(X\) is said to have the Klee property if there are two vector topologies on \(X\), say \({\mathcal T}_1\) and \({\mathcal T}_2\), such that the quasi-norm topology is the supremum of \({\mathcal T}_1\) and \({\mathcal T}_2\), such that \((X,{\mathcal T}_1)\) has trivial dual while the Hausdorff quotient of \((X,{\mathcal T}_2)\) is nearly convex. Klee raised the question of whether every topological vector space has the Klee property. Klee’s question has a negative answer [the first author, Stud. Math. 116, No. 2, 167-187 (1995)]. The aim of this paper is to characterize the class of separable quasi-Banach spaces with the Klee property. Using this characterization a counter-example to Klee’s question is given. If \(X\) is a quasi-Banach space with dual \(X^*\) the kernel of \(X\) is the linear subspace \(\{x: x^*(x)=0\), \(\forall x^*\in X^*\}\).
The main result of this paper is the following: Let \(X\) be a separable quasi-Banach space, with kernel \(E\). Then \(X\) fails to have the Klee property if and only if \(E\) has infinite codimension and the quotient map \(\pi:X\to X/E\) is strictly singular.

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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