Definition:Internal Hilbert Space Direct Sum

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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.

Also see