The uniform boundedness principle for bornological vector spaces (Q1319594)

The uniform boundedness principle is the following property of a function space \(L(E,F)\): a subset which is pointwise bounded is bounded on bounded sets. In this paper \(E\) and \(F\) are not locally convex spaces as usual, but bornological vector spaces, \(L(E,F)\) being formed by the bornological linear maps \(E\to F\). In this situation a much better theorem holds than in the classical case: six properties are given for \(E\), any of which being equivalent with the validity of the uniform boundedness principle for \(L(E,F)\) for all suitable \(F\). These conditions are satisfied in particular if \(E\) is a Mackey-complete convex bornological space. The stated results are crucial in modern differentiation theory; they imply e.g. that the smooth structure of \(L(E,F)\) is initial with respect to the point-evaluations.