# direct sum of Hilbert spaces

Let $\{H_{i}\}_{i\in I}$ be a family of Hilbert spaces indexed by a set $I$. The direct sum of this family of Hilbert spaces, denoted as

 $\bigoplus_{i\in I}H_{i}$

consists of all elements $v$ of the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) of $\{H_{i}\}_{i\in I}$ such that $\sum\|v_{i}\|^{2}<\infty$. Of course, for the previous sum to be finite only at most a countable number of $v_{i}$ can be non-zero.

Vector addition and scalar multiplication are defined termwise: If $u,v\in\bigoplus_{i\in I}H_{i}$, then $(u+v)_{i}=u_{i}+v_{i}$ and $(sv)_{i}=sv_{i}$.

The inner product of two vectors is defined as

 $\langle u,v\rangle=\sum_{i\in I}\langle u_{i},v_{i}\rangle$