\begin{definition}[Definition:Internal Hilbert Space Direct Sum]
Let $\HH$ be a Hilbert space.
Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed set of pairwise orthogonal subspaces of $\HH$.
Then the '''internal (Hilbert space) direct sum''' of the $M_i$ is their closed linear span $\vee_i M_i$.
It is denoted by $\bigoplus_i M_i$, or $\ds \bigoplus_{i \mathop \in I} M_i$ if the set $I$ is to be stressed.
When $I$ is finite, by Closed Linear Subspaces Closed under Setwise Addition, have that:
:$\ds \bigoplus_{i \mathop \in I} M_i = \sum_{i \mathop \in I} M_i$, where $\ds \sum$ signifies setwise addition.
\end{definition}