Topological structure of the family of solutions of the Bogolyubov equation

The Bogoliubov equation for generating functionals of classical statistical physics is studied in a specially constructed Banach space $H_\alpha$ of analytic functionals. Properties of closedness, compactness and convexity of the family of solutions of this equation in various topologies of the space $H_\alpha$ are considered. On the basis of the results obtained, the problem of the transition to the thermodynamic limit in the generating functional method is solved.