Definition:Hilbert Space Direct Sum

General Definition
Let $\left({H_i}\right)_{i \in I}$ be a $I$-indexed family of Hilbert spaces.

Let $\displaystyle \bigoplus_{i\in I} H_i = \left\{{h \in \prod_{i \in I} H_i: \sum \left\{{\left\Vert{h \left({i}\right)}\right\Vert^2: i \in I}\right\} < \infty}\right\}$, where $\prod$ denotes Cartesian product of sets, and $\sum$ denotes a generalized sum.

Define an inner product $\displaystyle \left\langle{\cdot, \cdot}\right\rangle$ on $\displaystyle \bigoplus_{i\in I} H_i$ by:


 * $\displaystyle \left\langle{g, h}\right\rangle = \sum \left\{{ \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle: i \in I}\right\}$

With respect to this inner product, $\displaystyle \bigoplus_{i\in I} H_i$ is a Hilbert space.

It is called the Hilbert space direct sum of the $H_i$, and is also denoted $\displaystyle \bigoplus \left\{{H_i: i \in I}\right\}$.

That it is indeed a Hilbert space (and that $\left\langle{\cdot, \cdot}\right\rangle$ is an inner product) is proved on Hilbert Space Direct Sum is Hilbert Space.