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 Quote by tom.stoer I think my example with S³ was missleading; I'll try it again: The dimension of the vector space of C-functions on S³ is infinite-dimensional. But if we look at the Lie group SU(2) we can write its elements as $$U[f] = \exp\left\{i t^a f^a(x)\right\}$$ The functions fa are members of an infinite dimensional vector space, but nevertheless we talk about a finite dimensional group b/c we have finitly many generators ta.
 Looking at the Ashtekar formulation of gravity there are the Gauss-law generators Ga(x) and the spacelike diffeomorphism generators = vector-constraints Va(x). The transformations are generated via functionals like $$G[f] = \int d^3x\,G^a(x)\,f(x);\quad V[f] = \int d^3x\,V^a(x)\,f(x);\quad$$ Again we have finitly many generators even if the functions f belong to an infinite dimensional vector space.