This issue is devoted to two issues: the study of the most important class of linear topological spaces - nuclear spaces and equipped with Hilbert spaces, and the study of harmonic analysis in Euclidean and infinite-dimensional linear spaces. Applications to the spectral analysis of linear operators, to the theory of measure in linear topological spaces 2, commutation relations in quantum field theory, generalized random processes, and t. are considered. d. Harmonic analysis on the Lorentz group and related questions of integral geometry will be presented in the fifth issue. The reader is expected to become acquainted with the first two chapters of the . 1. The necessary information from the second issue is summarized in this book .. The book is designed for undergraduate, graduate and research students.