Category:Hilbert Space Direct Sums

This category contains results about Hilbert space direct sums.
Definitions specific to this category can be found in Definitions/Hilbert Space Direct Sums.

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.