The Lie group of real analytic diffeomorphisms is not real analytic

What is it about?

Given an ordinary compact real analytic manifold, on can consider the group of all real analytic self-diffeomorphisms of it. This group has an infinite dimensional manifold structure in a natural way. However, the group operations are not real analytic, it is just a smooth Lie group.

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Why is it important?

This result is remarkable since you would not expect this sort of behaviour. The group operations are infact "conveniently analytic", so this example shows again the differences between the calculus used in this article and the convenient setting.