Orthonormal Basis and Hamel Basis Cardinality

Question

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but orthonormal basis in seperable spaces is countable.)

Answer 1

I didn't say "at least c", btw. Say $card(S)=c$. Then $S$ has exactly $c$ countable subsets: $c$ choices for the first element, $c$ choices for the second, etc, for a total of $c^{\aleph_0}=c$. Given a countable subset, how many ways are there to assign numbers to make an element of $\ell^2(S)$? There are $c$ choices for the first coordinate, etc, again it comes out to $c$. So $card(\ell^2(S)=c$. Now a Hamel basis is a subset of $\ell^2(S)$, so the cardinality of a Hamel basis is no more than $c$. But the cardinality of a Hamel basis is at least as large as the cardinality of an orthonormal basis. So the cardinality of a Hamel basis is $c$.