Patrick Cabau is an independent researcher. He received his M.Sc. at the University of Toulouse in 1980, l'Agrégation de Mathématiques in 1987 and his Ph.D. from the University of Savoy in 1999. He has taught at the ENSET (University Nord-Madagascar), has been a member of the LIM (Tunisia Polytechnic School) and a teacher in high schools. His primary current interests are in the areas of differential geometry, mechanics and mathematical physics.

Fernand Pelletier began his career as a researcher at the University of Burgundy in 1970. He obtained his third cycle doctorate in 1973 and his habilitation in 1980 at this University. Appointed Professor at the University of Corsica Pasquale Paoli in 1983, he was transferred to the University of Savoy in 1986. He was the Director of the Mathematics Laboratory (LAMA) of the University of Savoy from 1989 to 1996, Director of differential geometry teams in LAMA from 1996 to 2002 and Director of the "South Rodhanian" research group from 2002 to 2006. From his retirement in 2010 until now, he has been Professor Emeritus at the University of Savoy. His main research topics relate to differential geometry, dynamic systems, control theory and mathematical physics in finite and infinite dimensions.

This book describes in detail the basic context of the Banach setting and the most important Lie structures found in finite dimension. The authors expose these concepts in the convenient framework which is a common context for projective and direct limits of Banach structures.

1. Preliminaries. 2. Banach Lie structures. 3. Convenient structures. 4. Projective limits. 5. Direct limits. 6. Convenient Lie algebroids and prolongations. 7. Partial Poisson structures. 8. Integrability of distributions.