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This text is devoted to an account of one of the branches of functional analysis, the theory of commutative normed rings, and the principal applications of that theory. It is based on a paper written by the authors in 1940. The book consists of three parts. Part one is concerned with the theory of commutative normed rings and is divided into two chapters - the first containing foundations of the theory and the second dealing with more special problems. Part two deals with applications to harmonic analysis and is divided into three chapters - the first discusses the ring of absolutely...