Note on quasi-bounded sets (Q1175446)

The authors consider an example of two quasi-bounded sets such that their union and their closure both are not quasi-bounded. The example is based upon an observation that the spaces \(E_{B+f(B)}\) and \(E_{B\cup f(B)}\) both are not Hausdorff for a closed unit ball \(B\) in an infinite- dimensional Banach space \(X\) and an appropriate topological isomorphism \(f: E_ B\to E_{f(B)}\), where \(E_ A\) denotes, as usual, a locally convex space which is the linear hull of \(A\) equipped with the topology generated by the gauge of the absolutely convex hull of \(A\). Some incorrect propositions of the note \textit{J. Kučera} [Int. J. Math. Math. Sci. 13, No. 3, 607--610 (1990; Zbl 0745.46006)] are reformulated with regard to the example.