Definition:Partition of Unity (Hilbert Space)

Definition
Let $H$ be a Hilbert space.

A partition of unity or partition of identity on $H$ is a family $\left({P_i}\right)_{i \in I}$ of projections, subject to:


 * If $i \ne j$, then $P_i P_j = P_j P_i = 0$
 * $\vee \left\{{\operatorname{ran} P_i: i \in I}\right\} = H$, where $\vee$ signifies closed linear span

One may encounter the notations $1 = \sum_i P_i$ and $1 = \bigoplus_i P_i$.

Here, $1$ signifies the identity operator on $H$.