Analyticity of Gaussian measures

The notions of differentiability and analyticity of a (generalized) random function are introduced. It is proved that Gaussian random function is both infinitely differentiable and analytic (i. e., may be expanded in the power series). As an application of these results we prove that a bounded uniformly continuous functional defined on a subset of Frechet space may be uniformly approximated by analytical functionals. The analyticity of the fundamental solution of infinite-dimensional heat equation is proved also.