# 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

$$\underset{i\in I}{\oplus}{H}_{i}$$ |

consists of all elements $v$ of the Cartesian product^{} (http://planetmath.org/GeneralizedCartesianProduct) of ${\{{H}_{i}\}}_{i\in I}$ such that $$. 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 $u,v\in {\oplus}_{i\in I}{H}_{i}$, then ${(u+v)}_{i}={u}_{i}+{v}_{i}$ and ${(sv)}_{i}=s{v}_{i}$.

The inner product^{} of two vectors is defined as

$$\u27e8u,v\u27e9=\sum _{i\in I}\u27e8{u}_{i},{v}_{i}\u27e9$$ |

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 |