\begin{definition}[Definition:Hilbert Space Direct Sum]
Let $\family {H_i}_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces.
Let:
:$\ds \bigoplus_{i \mathop \in I} H_i = \set {h \in \prod_{i \mathop \in I} H_i: \sum \set {\norm {\map h i}_{H_i}^2: i \in I} < \infty}$
where:
:$\prod$ denotes Cartesian product of sets
:$\sum$ denotes a generalized sum.
Define an inner product $\innerprod \cdot \cdot$ on $\ds \bigoplus_{i \mathop \in I} H_i$ as:
:$\ds \innerprod g h = \sum \set {\innerprod {\map g i} {\map h i}_{H_i}: i \in I}$
With respect to this inner product, $\ds \bigoplus_{i \mathop \in I} H_i$ is a Hilbert space.
It is called the '''Hilbert space direct sum''' of the $H_i$, and is also denoted $\ds \bigoplus \set {H_i: i \in I}$.
That it is indeed a Hilbert space (and that $\innerprod \cdot \cdot$ is an inner product) is proved on Hilbert Space Direct Sum is Hilbert Space.
{{rewrite|Reformulate when Direct Sum of Vector Spaces is properly defined}}
\end{definition}