Function spaces and boundary value problems in domains of Hilbert space

We study the properties of function spaces on an infinite-dimensional Hilbert space that have derivatives of arbitrary order along basis directions summable with a certain weight. A nonnegative translation-invariant measure is introduced on the Hilbert space, serving as an analog of the Lebesgue measure on a finite-dimensional Euclidean space. Analogs of the space of smooth functions, Sobolev spaces, are introduced. Embedding theorems and trace theorems for functions in Sobolev spaces are established. Applications of the function spaces under study to differential equations and boundary value problems are explored. For infinite-dimensional domains, the Dirichlet problem for the Poisson equation is posed. Using the variational method, the existence and uniqueness of its solution are established.