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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 $I$ 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 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 $(sv)_i = s v_i$

The inner product of two vectors is defined as $$\langle u, v \rangle = \sum_{i \in I} \langle u_i, v_i \rangle$$




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See Also: category of Hilbert spaces


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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
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This is version 4 of direct sum of Hilbert spaces, born on 2004-10-12, modified 2008-10-19.
Object id is 6363, canonical name is DirectSumOfHilbertSpaces.
Accessed 2175 times total.

Classification:
AMS MSC46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )

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