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My question:

why don't we get a Hamel basis (a maximal linearly independent set) instead of a maximal orthonormal set for a Hilbert space. In what dimension can we use a Hamel basis and in which we can't?

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    These are different concepts: Linear Basis: $v=\alpha_1b_1+\ldots\alpha_n b_n$ Approximate Linear Basis: $v\approx\lambda_1e_1+\ldots+\lambda_n e_n$2015-02-23

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We know the existence of a dense countable orthonormal set i.e. $\fbox{1}\langle x,e_i\rangle \,$ for $\forall i \in \mathbb{N}$ $\, \Rightarrow x=0.\,$
Since a Hamel basis on a Hilbert space is uncountable we can extend the linearly independent set $ \{ e_i\ | \forall i \in \mathbb{N} \}\,$ to a Hamel basis in a non trivial way. So $\fbox{1} \,$ yields a contradiction.