Definition:Partition of Unity (Hilbert Space)

This page is about Partition of Unity in the context of Hilbert Space. For other uses, see Partition of Unity.

Definition

Let $H$ be a Hilbert space.

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

If $i \ne j$, then $P_i P_j = P_j P_i = 0$

$\vee \set {\Img {P_i}: i \in I} = H$, 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 $H$.