Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group \(G\) can be approximated by Lie groups in the sense that every identity neighborhood \(U\) of \(G\) contains a normal subgroup \(N\) such that \(G/N\) is a Lie group, then it is called a proLie group. Every locally compact connected topological group and every compact group is a proLie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of proLie groups is. For half a century, locally compact proLie groups have drifted through the literature, yet this is the first book which systematically treats the Lie and structure theory of proLie groups irrespective of local compactness. This study fits very well into the current trend which addresses infinitedimensional Lie groups. The results of this text are based on a theory of proLie algebras which parallels the structure theory of finitedimensional real Lie algebras to an astonishing degree, even though it has had to overcome greater technical obstacles. This book exposes a Lie theory of connected proLie groups (and hence of connected locally compact groups) and illuminates the manifold ways in which their structure theory reduces to that of compact groups on the one hand and of finitedimensional Lie groups on the other. It is a continuation of the authors' fundamental monograph on the structure of compact groups (1998, 2006) and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis, and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of current research, which has so many fruitful interactions with other fields of mathematics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Advanced graduate students interested in proLie groups. Table of Contents  Panoramic overview
 Limits of topological groups
 Lie Groups and the Lie theory of topological groups
 ProLie groups
 Quotients of proLie groups
 Abelian proLie groups
 Lie's third fundamental theorem
 Profinitedimensional modules and Lie algebras
 The structure of simply connected proLie groups
 Analytic subgroups and the Lie theory of proLie groups
 The global structure of connected proLie groups
 Splitting theorems for proLie groups
 Compact subgroups of proLie groups
 Iwasawa's local splitting theorem
 Catalog of examples
 Appendix 1. The CampbellHausdorff formalism
 Appendix 2. Weakly complete topological vector spaces
 Appendix 3. Various pieces of information on semisimple Lie algebras
 Bibliography
 List of symbols
 Index
