Counterexample on products of bornological spaces (Q1590720)

A linear topological space \(X\) is called bornological and \({\mathcal L}\)-bornological if every bounded linear map on \(X\) with values in a locally convex space and a linear topological space, respectively, is continuous. It is known [\textit{G. W. Mackey}, Bull. Am. Math. Soc. 50, 719-722 (1944; Zbl 0060.13402)] that the product of Ulam nonmeasurably many bornological spaces is also bornological. \textit{N. Adash} [Arch. Math. 61, No. 1, 88-89 (1993; Zbl 0818.46003)] and \textit{A. P. Robertson} [Glasg. Math. J. 11, 37-40 (1970; Zbl 0193.40702)] proved similar results for \({\mathcal L}\)-bornological spaces. \textit{H. Pfister} [``Räume von stetigen Funktionen und summenstetige Abbildungen.'' Habilitationsschrift, München, 1978] was aware of a gap in Adash's proof and remarked that the result holds under the continuum hypothesis. In the present paper, under a consistent set-theoretical assumption, the author gives a counterexample to the results of Adash and Robertson and remarks that these results ``hold (excluding trivial cases) exactly for products indexed by cardinals smaller than the first sequential cardinal''.