This interesting answer raised a question in my mind:
Suppose that $M$ and $N$ are smooth manifolds and $f:$ $M \rightarrow N$ is a diffeomorphism. Can we conclude that $C^{\infty} (M)$ and $C^{\infty} (N)$ are isomorphic? In other words, is there a one to one correspondence between smooth functions on the two smooth manifolds? ($C^{\infty} (M)$ denotes the set of smooth functions from $M$ to $\mathbf{R}$).
Thanks