The Mackey- Ulam theorem in topological vector spaces
A topological vector space \(E\) is called \({\mathcal L}\)-bornological, if every bounded linear map from \(E\) into any topological vector space \(F\) is continuous. The author proves that a product \(\prod_{\alpha\in I} E_ \alpha\) of arbitrary \({\mathcal L}\)-bornological spaces is again \({\mathcal L}\)-bornological iff the locally convex space \(\omega(I)\) of all functions from \(I\) to \(\mathbb{R}\) (or \(\mathbb{C}\)) is bornological in the locally convex sense. This is the case if the cardinality of the index set \(I\) is smaller than the smallest strongly non accessible cardinal number [see \S28 (8) in \textit{G. Köthe}: `Topologische lineare Räume. I' (1966; Zbl 0137.313)].
scientific article; zbMATH DE number 446298