In this revised and extended version of his course notes from a 1-year
course at Scuola Normale Superiore, Pisa, the author provides an
introduction - for an audience knowing basic functional analysis and
measure theory but not necessarily probability theory - to analysis in a
separable Hilbert space of infinite dimension. Moreover, some details
have been added as well as some new material on dynamical systems with
dissipative nonlinearities and asymptotic behavior for gradient systems.