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Since the above answers and comments refer to the concept of differentiability for functions not just with values IN an infinitely dimensional space, but also defined ON one, despite the fact that your question addresses functions defined on a Lie group and so a finite dimensional space and you explicitly state that this reduce to the case of functions on euclidean space by localisation, let me take the liberty of adding some information on this situation. This was considered in some detail in the article "Espaces de fonctions différentiables a valeurs vectorielles", Jour. d'Anal. 4 (1954-56) 88-148 by Laurent Schwartz which is probably still the definitive statement on the subject. The definition is exactly as in the scalar case using limits of difference quotients. Schwartz uses the theory of nuclear spaces and tensor products which had just been invented by Grothendieck for the somewhat analogous case of holomorphic functions and the fact that the smooth functions form a nuclear space makes the theory somewhat simpler.