PlanetMath: direct sum of Hilbert spaces

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direct sum of Hilbert spaces

(Definition)

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

$\displaystyle \bigoplus_{i \in I} H_i$

consists of all elements

of the Cartesian product of $\{H_i\}_{i \in I}$ such that $\sum \Vert v_i\Vert^2 < \infty$ . Of course, for the previous sum to be finite only at most a countable number of

can be non-zero.

Vector addition and scalar multiplication are defined termwise: If $u, v \in \bigoplus_{i \in I} H_i$ , then and .

The inner product of two vectors is defined as

$\displaystyle \langle u, v \rangle = \sum_{i \in I} \langle u_i, v_i \rangle$

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"direct sum of Hilbert spaces" is owned by asteroid. [ full author list (2) | owner history (2) ]

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Attachments:: direct sum of bounded operators on Hilbert spaces (Definition) by asteroid

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Cross-references: vectors, inner product, multiplication, scalar, vector addition, number, countable, finite, sum, direct sum, indexed by, Hilbert spaces
There are 3 references to this entry.

This is version 3 of direct sum of Hilbert spaces, born on 2004-10-12, modified 2008-04-09.
Object id is 6363, canonical name is DirectSumOfHilbertSpaces.
Accessed 1420 times total.

Classification:

AMS MSC:

46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )

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