Mathematical description of the equilibrium state of classical systems on the basis of the canonical ensemble formalism

This paper gives a rigorous mathematical description of the equilibrium state of the infinite system of particles on the basis of canonical ensemble theory. A proof is given of the existence and uniqueness of the limiting distribution functions and their analytical dependence on density. Results have been obtained by using methods developed by two of the authors in 1949, and based on the application of the theory of Banaeh spaces to the study of the equation forthe distribution functions.