On a theorem of Laurent Schwartz (Q627256)

Laurent Schwartz (1966) proved that, for a large class of topological vector spaces \(E\) and \(F\), every linear transformation \(E\to F\) whose graph is a Borel set is continuous (technically, \(E\) needs to be locally convex and ultra-bornological, \(F\) a locally convex Souslin space). In this note, the author gives a short and very different proof of a similar result: if \(E\) and \(F\) are real Banach spaces and if the graph of a linear map \(\psi:E\to F\) is measurable with respect to every centered Gaussian measure on \(E\), then \(\psi\) is continuous. As an argument by Gilles Pisier shows (as quoted at the end of the article), if \(\psi\) is Borel measurable, then this theorem follows immediately from Schwartz' theorem. Conversely, if \(E\) and \(F\) are separable, then Schwartz' theorem follows from Stroock's result because Borel measurability of the graph of \(\psi\) then implies that measurability with respect to centered Gaussian maps is assured.