Continuity of universally measurable linear maps (Q1056864)

The paper goes up to the well-known Banach theorem asserting that every linear map F from a Banach space X into a normed space Y, which is measurable (with respect to the Borel \(\sigma\)-algebra of X) must be continuous. In 1966 L. Schwartz has strengthened Banach's theorem. He proved that if F is \(\mu\)-measurable for any Radon measure \(\mu\) on X, then F is continuous (in Schwartz's theorem X is an ultrabornological locally convex space, and Y is an arbitrary locally convex space). The paper under review deals with a further strengthening of the Banach theorem. The main assertion of the paper is as follows. Theorem. Let X and Y be locally convex topological vector spaces, with X sequentially complete, and let F:X\(\to Y\) be a linear map. The following statements are equivalent: (1) F is \(\mu\)-measurable for any Gaussian measure \(\mu\) on X which is a linear image of a Gaussian measure on a separable Hilbert space. (2) F is bounded (i.e. the image of any bounded set is bounded). The author shows that the Schwartz's theorem is an easy corollary of this theorem. The author proves also variants of the above Theorem in the non- locally convex case.