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[parent] all orthonormal bases have the same cardinality (Theorem)

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

$ \,$

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.

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$

$\displaystyle \Vert x\Vert^2 = \sum_{i \in I} \vert\langle x, e_i \rangle\vert^2 $
We know that, in the above sum, $ \langle x, e_i \rangle \neq 0$ for only a countable number of $ i \in I$. Thus, considering $ x$ as $ f_j$, the set $ I_j := \{ i \in I: \langle f_j, e_i \rangle \neq 0 \}$ is countable. Since for each $ i \in I$ we also have
$\displaystyle \Vert e_i\Vert^2 = \sum_{j \in J}\vert \langle e_i, f_j \rangle\vert^2 $
there must be $ j \in J$ such that $ \langle f_j, e_i \rangle \neq 0$. We conclude that $ \displaystyle I = \bigcup_{j \in J} I_j$.

Hence, since each $ I_j$ is countable, $ I \leq J\!\times\!\mathbb{N} \cong J$ (because $ J$ is infinite).

An analogous argument proves that $ J \leq I$. Hence, by the Schroeder-Bernstein theorem $ J$ and $ I$ have the same cardinality. $ \square$



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Other names:  dimension of an Hilbert space is well-defined

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Cross-references: Schroeder-Bernstein theorem, countable, sum, Parseval's equality, infinite, infinite dimensional, indexed by, infinite-dimensional, Hamel basis, orthonormal basis, vector space, finite-dimensional, cardinality, Hilbert space, bases, orthonormal
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This is version 2 of all orthonormal bases have the same cardinality, born on 2008-03-21, modified 2008-03-22.
Object id is 10433, canonical name is AllOrthonormalBasisHaveTheSameCardinality.
Accessed 183 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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