# all orthonormal bases have the same cardinality

Theorem^{}. – All orthonormal bases of an Hilbert space^{} $H$ have the same cardinality. It follows that the concept of dimension^{} of a Hilbert space is well-defined.

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*Proof:* When $H$ is finite-dimensional (as a vector space^{}), every orthonormal basis is a Hamel basis^{} of $H$. Thus, the result follows from the fact that all Hamel bases of a vector space have the same cardinality (see this entry (http://planetmath.org/AllBasesForAVectorSpaceHaveTheSameCardinality)).

We now consider the case where $H$ is infinite-dimensional (as a vector space). Let ${\{{e}_{i}\}}_{i\in I}$ and ${\{{f}_{j}\}}_{j\in J}$ be two orthonormal basis of $H$, indexed by the sets $I$ and $J$, respectively. Since $H$ is infinite dimensional the sets $I$ and $J$ must be infinite^{}.

We know, from Parseval’s equality, that for every $x\in H$

$${\parallel x\parallel}^{2}=\sum _{i\in I}{|\u27e8x,{e}_{i}\u27e9|}^{2}$$ |

We know that, in the above sum, $\u27e8x,{e}_{i}\u27e9\ne 0$ for only a countable^{} number of $i\in I$. Thus, considering $x$ as ${f}_{j}$, the set ${I}_{j}:=\{i\in I:\u27e8{f}_{j},{e}_{i}\u27e9\ne 0\}$ is countable. Since for each $i\in I$ we also have

$${\parallel {e}_{i}\parallel}^{2}=\sum _{j\in J}{|\u27e8{e}_{i},{f}_{j}\u27e9|}^{2}$$ |

there must be $j\in J$ such that $\u27e8{f}_{j},{e}_{i}\u27e9\ne 0$. We conclude that $I={\displaystyle \bigcup _{j\in J}}{I}_{j}$.

Hence, since each ${I}_{j}$ is countable, $I\le J\times \mathbb{N}\cong J$ (because $J$ is infinite).

An analogous proves that $J\le I$. Hence, by the Schroeder-Bernstein theorem $J$ and $I$ have the same cardinality. $\mathrm{\square}$

Title | all orthonormal bases have the same cardinality |
---|---|

Canonical name | AllOrthonormalBasesHaveTheSameCardinality |

Date of creation | 2013-03-22 17:56:10 |

Last modified on | 2013-03-22 17:56:10 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 7 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46C05 |

Synonym | dimension of an Hilbert space is well-defined |