Lemma 11.10.2.A2189A9C7D5A Let $M$ be a compact smooth manifold and let $\mathcal S$ be a sheaf of germs of continuous real functions on $M$, whose elements we call sections. Suppose that $\mathcal S$ is closed under maxima and has the following two properties.
Stability: if $f$ is a section near a compact set $K$, then $f+u$ is a section near $K$ whenever $u$ is smooth and sufficiently small in the $C^\infty $ topology on a neighborhood of $K$.
Local approximation: every point has a neighborhood $U$ such that, for every compact $K\Subset U$, a section near $K$ can be approximated uniformly on $K$ by smooth sections near $K$, with arbitrarily small errors in any prescribed finite collection of derivative norms on compact subsets where the section is already smooth.
Then every global section can be approximated uniformly on $M$ by smooth global sections.
Proof.
We first describe a gluing step. Let $f$ be a global section that is smooth near a compact set $L_1$, and let $L_2$ be a compact subset of one of the neighborhoods $U$ in the statement. Choose relatively compact open sets $W_3,W_4$ with $L_1\subset W_3$, $f$ smooth near $\overline W_3$, and $L_2\subset W_4\Subset U$. Choose a smooth function $0\leq \rho \leq 1$ equal to one near $\overline W_3$, with compact support in the region where $f$ is smooth. Put $V=W_3\cup W_4$, and choose a smooth function $-1\leq \psi \leq 1$ equal to one near $L_1\cup L_2$ and equal to minus one near $M\setminus V$.
By stability, for sufficiently small $\xi >0$ the function $f_\xi =f+\xi \psi $ is still a global section. Approximate $f_\xi $ on $\overline W_4$ by a smooth section $g$, with uniform error less than $\delta <\xi /2$, and with arbitrarily small derivative errors near the compact set $\overline W_4\cap \operatorname {supp}\rho $, where $f_\xi $ is smooth. On $V$ define
\[ h=f_\xi \quad \hbox{on }W_3,\qquad h=\rho f_\xi +(1-\rho )g\quad \hbox{on }W_4. \]
These definitions agree on the overlap, and $h$ is smooth. Where $\rho $ vanishes, $h$ is the section $g$. Where $\rho $ can be nonzero, $h-f_\xi =(1-\rho )(g-f_\xi )$, and stability applied to the smooth section $f_\xi $ makes $h$ a section if the derivative errors of $g-f_\xi $ are sufficiently small. Moreover, $|h-f_\xi |<\delta $.
Define $f_1=\max \{ f,h\} $ on $V$ and $f_1=f$ outside $V$. Near $L_1\cup L_2$ we have $h>f$, and near the boundary of $V$ we have $h<f$. Thus $f_1$ is continuous, it is a section because $\mathcal S$ is closed under maxima, and it is smooth near $L_1\cup L_2$. Also $\| f_1-f\| _{C^0}<\xi +\delta $. Near $L_1$ it equals $f+\xi $, so the previous smoothness is preserved.
Now cover $M$ by finitely many compact sets $L_2$ of this kind and repeat the step, starting with $L_1$ empty and then taking for $L_1$ the union of the previously treated compact sets. At every step choose $\xi $ and $\delta $ so small that the sum of the finitely many uniform errors is below the prescribed tolerance. The final section is smooth everywhere and has the required uniform error.
$\square$
Pome
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