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Let $M$ be a smooth manifold and $C^{\infty}\left(M\right)$ be the space of all smooth functions on $M$. My question is

(a) What is the standard smooth structure that makes $C^{\infty}\left(M\right)$ a smooth manifold? That is, how can we define coordinate charts on $C^{\infty}\left(M\right)$?

(b) Let $f\in C^{\infty}\left(M\right)$, what is the tangent space $T_{f}\left(C^{\infty}\left(M\right)\right)$?

Thank you.

Binjiu
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    It's a vector space. If you believe in manifolds modeled on particularly general topological vector spaces, like Frechet spaces, then the identity map is a perfectly good chart. If you believe only in finite dimensional manifolds, you're out of luck. –  Oct 05 '16 at 01:00
  • I don't understand your answer. Can you mention more clearly about my question? I suppose $M$ has a finite dimension. – Binjiu Oct 05 '16 at 01:14
  • $C^\infty(M)$ is not even locally compact. It is not a finite dimensional manifold. Did you want it to be? –  Oct 05 '16 at 01:35
  • I know that $C^{\infty}\left(M\right)$ is not a finite dimensional manifold, but my question is "is it a smooth manifold in some sense?" I just know about finite dimensional manifold. – Binjiu Oct 05 '16 at 03:59

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