Squared partition of unity

A squared partition of unity on an open set $U$ is a family of smooth real functions $(\chi_j)$ , with locally finite supports, such that

\sum_j\chi_j(x)^2=1\qquad(x\in U).

It is subordinate to an open cover $(U_j)$ when $\operatorname{supp}\chi_j\subseteq U_j$ .

Smooth normalization

Start with a locally finite family of smooth real functions $(\psi_j)$ having the desired supports and with at least one nonzero at every point. Then

\chi_j=\frac{\psi_j}{\sqrt{\sum_i\psi_i^2}}

is a squared partition. The denominator is strictly positive and smooth, because the sum is locally finite and the square-root function is smooth on $(0,\infty)$ . Supports are preserved.

Relation to an ordinary partition

The functions $\chi_j^2$ give a partition of unity. Taking the square root of each member of an arbitrary smooth nonnegative partition is not the same construction: for instance, $\sqrt{x^2}=|x|$ fails to be smooth at zero.

References

Richard Schwartz, Partitions of Unity (Brown lecture notes).