As I understand it, for the Lie group of positive real numbers under multiplication, the exp map from its Lie algebra as a vector space of all real numbers to this group is a global diffeomorphism.
I am tempted to think that the same conclusion would hold for the infinite-dimensional Lie group (in the IHL sense, if appropriate) of positive real-valued functions on a (compact) manifold under multiplication whose Lie algebra is the vector space of all real-valued functions. Assume proper Sobolev spaces are used, just for the peace of mind.
Is this true? If so, how to see it; if not, what goes wrong?
Thanks.