Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an epsilon surrounding of $M$, such that those $\alpha_i$ form a partition of unity on M?

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    $\begingroup$ What do you mean by partition of unity for functions whose values are in $X$? $\endgroup$ – user38355 Dec 16 '13 at 14:30
  • $\begingroup$ that means that there sum at every point on $M$ is always equal to one. $\endgroup$ – user66906 Dec 16 '13 at 14:32
  • $\begingroup$ Right, but then your functions would have to take real values, not values in $X$. $\endgroup$ – user38355 Dec 16 '13 at 14:33
  • $\begingroup$ Even with real values, there is something odd here. It seems that you can use just a single $a_i$ which is $1$ on $M$ and has support in $N$. If nothing more is desired, the problems appears trivial. $\endgroup$ – Harald Hanche-Olsen Dec 16 '13 at 14:40
  • $\begingroup$ it is not $C^{\infty}$ on X $\endgroup$ – user66906 Dec 16 '13 at 14:45

As Harald Hanche-Olsen said, one $\alpha_i$ is enough. Let $E$ be the $\epsilon/3$ neighborhood of $M$. Convolve $\chi_{E}$ with a $C^\infty$ bump function $\phi$ such that $$\int_{\mathbb R^n} \phi=1\quad \text{and}\quad \operatorname{supp}\phi\subset \{x:|x|<\epsilon/3\}$$ The convolution $\chi_E*\phi$ is $C^\infty$-smooth, is equal to $1$ on $M$, and its support is contained in $N$.


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