We know as "Partition of unity" that follow:
Let $X\subseteq \mathbb{R}^n$ be an open set, and let $K$ be a compact subset of $X$. Let $X_i$, $i=1,\ldots, m$, be open subsets of $X$ whose union contains $K$. Then one can find functions $\psi_1,\ldots, \psi_m$ such that $$\psi_i\in C_0^{\infty}(X_i),\quad 0\leq \psi_i\leq 1,\quad i=1,\ldots, m,$$ $$\sum_{i=1}^m\psi_i\leq 1\quad\mbox{on }X,\quad \sum_{i=1}^m\psi_i = 1\quad \mbox{on a neighbourhood of } K.$$
But I want to know if is possible to get a variant where $$\sum_{i=1}^m(\psi_i)^2 = 1\quad \mbox{on a neighbourhood of } K.$$ I tried to use the traditional Partition of Unity to create $\widetilde{\psi}_i$ such that $$\sum_{i=1}^m\widetilde{\psi}_i = 1\quad \mbox{on a neighbourhood of } K,$$ and then define $\psi_i := \sqrt{\widetilde{\psi}_i}$, but the problem is: How I can to verify that $\psi_i\in C_0^{\infty}(X_i)$? Because $\sqrt{\;\;}$ have problems in $x = 0$.
Please, Can somebody help me?
Thanks in advance.