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I want to show the characterizations of $C_p^{\infty}(\mathbb R^n)$ are equivalent:

$$A= \{[f] | \text{smooth} \ f: \mathbb R^n \to \mathbb R \}$$

$$B= \{[f] | \text{smooth} \ f: U_p \to \mathbb R \}$$

$$C= \{[f] | \text{smooth} \ f: V_{f,p} \to \mathbb R \}$$

where

  • all the germs in $B$ can be represented by functions whose domains are a fixed open subset $U_p$ of $\mathbb R^n$ that contains $p$

  • while the domaina of functions that represent germs in $C$ are any open subsets $V_{f,p}$ of $\mathbb R^n$ that contain $p$.

Note: I am attempting prove $A=B=C$ really as equal sets are not merely that they are in bijection.

So far I have done:

$(A \subseteq B):$

  • Let $[f] \in A$ for a smooth $f: \mathbb R^n \to \mathbb R$. We must show $[f] \in B$, which is done by finding some smooth $g: U_p \to \mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p \subseteq \mathbb R^n \cap U_p = U_p$ of $\mathbb R^n$ that contains $p$.

  • We can choose $g = f|_{U_p}$ and $W_p = U_p$.

$(B \subseteq C):$

  • Let $[f] \in B$ for a smooth $f: \mathbb U_p \to \mathbb R$. We must show $[f] \in C$, which is done by finding some smooth $g: V_{p,g} \to \mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p \subseteq V_{p,g} \cap U_p$ of $\mathbb R^n$ that contains $p$.

  • We can choose $V_{p,g} = U_p$ and $g = f$.

$(C \subseteq A):$

  • According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $\mathbb R^n$ for $n=1$, doesn't always exist. However, it exists to say $C \subseteq A$:

  • Let $[f] \in C$ for a smooth $f: \mathbb V_{p,f} \to \mathbb R$. We must show $[f] \in A$, which is done by finding some smooth $g: \mathbb R^n \to \mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p \subseteq V_{p,f} \cap \mathbb R^n = V_{p,f}$ of $\mathbb R^n$ that contains $p$.

  • We can choose $g = \rho f \ 1_{U_p}$, where $\rho: \mathbb R^n \to \mathbb R$ is a smooth bump function supported in $U_p$ and $\rho(D_p)=\{1\}$ for some open subset $D_p$ of $\mathbb R^n$ containing $p$, and we can choose $W_p=D_p$.

Questions

  1. Is anything wrong?

  2. Can $C \subseteq A$ be proven without bump functions?

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