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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.
Diffeomorphisms which are infinitesimally close to the identity look like gauge transformations. The problem is that infinitesimal diffeomorphisms do not generate the whole group.

 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.
If there are a finite number of generators, we are not talking about the full diffeomorphism group, only the infinitesimal diffeomorphisms.