Definition:Hilbert Space Direct Sum

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Definition

Let $\family {H_i}_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces.

Let:

$\ds \bigoplus_{i \mathop \in I} H_i = \set {h \in \prod_{i \mathop \in I} H_i: \sum \set {\norm {\map h i}_{H_i}^2: i \in I} < \infty}$

where:

$\prod$ denotes Cartesian product of sets
$\sum$ denotes a generalized sum.

Define an inner product $\innerprod \cdot \cdot$ on $\ds \bigoplus_{i \mathop \in I} H_i$ as:

$\ds \innerprod g h = \sum \set {\innerprod {\map g i} {\map h i}_{H_i}: i \in I}$

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

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


That it is indeed a Hilbert space (and that $\innerprod \cdot \cdot$ is an inner product) is proved on Hilbert Space Direct Sum is Hilbert Space.


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Direct Sum of Two Hilbert Spaces

Let $H, K$ be Hilbert spaces.

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

Define an inner product $\innerprod \cdot \cdot$ on $H \oplus K$ by:

$\innerprod {h_1 \oplus k_1} {h_2 \oplus k_2} = \innerprod {h_1} {h_2}_H + \innerprod {k_1} {k_2}_K$

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 $\sequence {H_n}_{n \mathop \in \N}$ be a sequence of Hilbert spaces.

Let $\ds \bigoplus_{n \mathop = 1}^\infty H_n = \set {\sequence {h_n}_{n \mathop \in \N}: h_n \in H_n, \sum_{n \mathop = 1}^\infty \norm {h_n}_{H_n}^2 < \infty}$.

Define an inner product $\innerprod \cdot \cdot$ on $\ds \bigoplus_{n \mathop = 1}^\infty H_n$ as:

$\ds \innerprod {\sequence {g_n}_{n \mathop \in \N} } {\sequence {h_n}_{n \mathop \in \N} } = \sum_{n \mathop = 1}^\infty \innerprod {g_n} {h_n}_{H_n}$

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


Sources

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