I'm working through a proof in my Differential Geometry text book and it uses a permutation (of the basis vectors) to construct a particular function. In order to prove this function is $C^k$, we have to know the permutation is a diffeomorphism. Is this true in general?
2026-03-26 19:03:44.1774551824
Are permutations diffeomorphisms?
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For sure. A permutation is a bijective linear map which is $\mathcal C^\infty$ (at least for finite dimensional linear spaces) and invertible.