Definition:Internal Hilbert Space Direct Sum
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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) $\text {II}.3.4$