direct sum of Hilbert spaces

Let $\{H_{i}\}_{i\in I}$ be a family of Hilbert spaces indexed by a set $I$ . The direct sum of this family of Hilbert spaces, denoted as

\bigoplus_{i\in I}H_{i}

consists of all elements $v$ of the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) of $\{H_{i}\}_{i\in I}$ such that $\sum\|v_{i}\|^{2}<\infty$ . Of course, for the previous sum to be finite only at most a countable number of $v_{i}$ can be non-zero.

Vector addition and scalar multiplication are defined termwise: If $u,v\in\bigoplus_{i\in I}H_{i}$ , then $(u+v)_{i}=u_{i}+v_{i}$ and $(sv)_{i}=sv_{i}$ .

The inner product of two vectors is defined as

\langle u,v\rangle=\sum_{i\in I}\langle u_{i},v_{i}\rangle

Linked PDF file:

http://images.planetmath.org/cache/objects/6363/pdf/DirectSumOfHilbertSpaces.pdf

Title	direct sum of Hilbert spaces
Canonical name	DirectSumOfHilbertSpaces
Date of creation	2013-03-22 14:43:55
Last modified on	2013-03-22 14:43:55
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	10
Author	asteroid (17536)
Entry type	Definition
Classification	msc 46C05
Related topic	CategoryOfHilbertSpaces