Definition:Hilbert Space Direct Sum

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General Definition

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

Let $\displaystyle \bigoplus_{i\in I} H_i = \left\{{h \in \prod_{i \in I} H_i: \sum \left\{{\left\Vert{h \left({i}\right)}\right\Vert_{H_i}^2: i \in I}\right\} < \infty}\right\}$, 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:

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

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.

Direct Sum of Two Hilbert Spaces

Let $H, K$ be Hilbert spaces.

Let $H \oplus K = \left\{{h \oplus k: h \in H, k \in K}\right\}$.

Define an inner product $\left\langle{\cdot, \cdot}\right\rangle$ on $H \oplus K$ by:

$\left\langle{h_1 \oplus k_1, h_2 \oplus k_2}\right\rangle = \left\langle{h_1, h_2}\right\rangle_H + \left\langle{k_1, k_2}\right\rangle_K$

With respect to this inner product, $H \oplus K$ is a Hilbert space called the Hilbert space direct sum of $H$ and $K$.


Direct Sum of Sequence of Hilbert Spaces

Let $\left({H_n}\right)_{n \in \N}$ be a sequence of Hilbert spaces.

Let $\displaystyle \bigoplus_{n=1}^\infty H_n = \left\{{ \left({h_n}\right)_{n \in \N}: h_n \in H_n, \sum_{n=1}^\infty \left\Vert{h_n}\right\Vert_{H_n}^2 < \infty}\right\}$.

Define an inner product $\displaystyle \left\langle{\cdot, \cdot}\right\rangle$ on $\displaystyle \bigoplus_{n=1}^\infty H_n$ by:

$\displaystyle \left\langle{ \left({g_n}\right)_{n \in \N}, \left({h_n}\right)_{n \in \N} }\right\rangle = \sum_{n=1}^\infty \left\langle{ g_n, h_n }\right\rangle_{H_n}$

With respect to this inner product, $\displaystyle \bigoplus_{n=1}^\infty H_n$ is a Hilbert space, called the Hilbert space direct sum of the $H_n$.


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