The Lie Group in Infinite Dimension

A Lie group acting on finite‐dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite‐dimensional case. We prove that the (local, C∞ smooth) action of a Lie group on infinite‐dimensional space (a manifold modelled on ℝ∞) may be regarded as a limit of finite‐dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher‐order) infinitesimal symmetries of differential equations.