The Lie Group in Infinite Dimension

A Lie group acting on finite-dimensional space is generated by its infinitesimaltransformations and conversely, any Lie algebra of vector fields in finite dimensiongenerates a Lie group (the first fundamental theorem). This classical result is adjustedfor the infinite-dimensional case. We prove that the (local, C∞ smooth) actionof a Lie group on infinite-dimensional space (a manifold modelled on ℝ∞) may beregarded as a limit of finite-dimensional approximations and the corresponding Liealgebra of vector fields may be characterized by certain finiteness requirements. Theresult is applied to the theory of generalized (or higher-order) infinitesimal symmetriesof differential equations