Like the title says, suppose $K\subset \mathbb{R}$ is a locally compact space without isolated points and $\left \{ U_i \right \}$, ${i \in \{1,\cdots ,n \}}$ is an open cover of $K$. Are There a partition of unity on $K$ subordinate to $\left \{ U_i \right \}$ consisting of functions $\{ f_{i}\}_{i \in \{1,\cdots,n \}}$ such that:
$$ f_i \in C_0^{1}(K,[0,1]) \quad \text{for all $i \in \{1,\cdots,n \}$}? $$
where $C_0^{1}(K,[0,1])$ is the banach space consisting of all continously differentiable functions $f$ such that $f, f'$ vanish at infinity (provided for example the max norm).
So far I've seen : Existence of a partition of unity with uniformly bounded derivatives
Is there any article, book that take partitions of unity with this properties? perhaps I just need to follow the main proof? Thanks in advanced.
No, such a partition of unity never exists if $K$ is not compact. Indeed, the sum $f_1+\dots+f_n$ would also vanish at infinity, but that is impossible since this sum must be equal to $1$ which does not vanish at infinity.
If you drop the requirement that the $f_i$ are indexed by the same set as the $U_i$, though, then such partitions of unity do exist. First, since $K$ is locally compact, you can refine your open cover to an open cover $\mathcal{V}$ by precompact open sets. Then you can take any smooth partition of unity subordinate to $\mathcal{V}$. Since each element of $\mathcal{V}$ is precompact, each function in this partition of unity has compact support in $K$ and so in particular it and its derivative vanishes at infinity.