A Hilbert space whose Hamel dimension and Hilbert dimension are different.

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The Hilbert space l^{2} has the orthonormal basis (e_{n}) with e_{n}(m) = δ_{mn}; m, n ∈ N. Hence its Hilbert dimension is ℵ_{0}. But the set of all sequences x_{α} =< 1, α, α^{2}, α^{3}, ⋯ >, 0 < α < 1 is a linearly independent uncountable subset of l^{2}. Thus the Hamel dimension of l^{2} isn’t ℵ_{0}.

Comment. This Hilbert dimension is probably the only one which this can happen.