Function Spaces and Boundary Value Problems in Domains of a Hilbert Space

We study the properties of spaces of functions on an infinite-dimensional Hilbert space whose derivatives of arbitrary order along basis directions exist and are integrable with a certain weight. To this end, we introduce a nonnegative translation-invariant measure on the Hilbert space which serves as an analog of the Lebesgue measure on finite-dimensional Euclidean space. We introduce analogs of the space of smooth functions and of the Sobolev spaces, and establish embedding and trace theorems for the latter. We also consider some applications of the function spaces under study to differential equations and boundary value problems. In particular, we pose a Dirichlet problem for the Poisson equation on an infinite-dimensional domain and apply the variational method in order to establish the existence and uniqueness of its solutions.