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 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 $(sv)_i = s v_i$

The inner product of two vectors is defined as $$\langle u, v \rangle = \sum_{i \in I} \langle u_i, v_i \rangle$$