This comprehensive monograph is devoted to a systematic treatment of pro-Lie groups. The notion of a pro-Lie group is a generalization of compact groups as well as of locally compact connected topological groups with a property that an arbitrary product of pro-Lie groups is again a pro-Lie group. The monograph is devoted to a systematic treatment of pro-Lie groups based on a theory of pro-Lie algebras, which is very much similar to the structure theory of finite dimensional Lie algebras. The book is very carefully organised. It begins with 60 pages of an overview of the contents of the book describing the structure of the theory. A good understanding of projective limits is a necessary prerequisite for the book, although a careful description of it can be found in chapter 1. Pro-Lie groups are introduced in chapter 3. A structure of commutative pro-Lie groups and their relations to weakly complete (infinite dimensional) topological vector spaces are discussed in chapter 5. As in the classical theory, a key point of the theory is a structure theory for pro-Lie algebras, introduced in chapter 7. The full structure theory is then treated in chapters 8–13. The book ends with a catalogue of examples. The book is very well written and it offers a first systematic treatment of the subject in the literature.

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