Definition:Internal Hilbert Space Direct Sum

Definition

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

• Definition:Hilbert Space Direct Sum, a broad generalization

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.3.4$