Every Hilbert space is determined up to a unitary isomorphism by the cardinality of its orthonormal basis (which exists by Gram-Schmidt). My question:
Let $\mathbb{K}$ be the field either $\mathbb{R}$ or $\mathbb{C}$. Define the algebraic dimension of an abstract vector space over $\mathbb{K}$ to be the cardinality of its Hamel basis over $\mathbb{K}$. Is a Hilbert space $\mathscr{H}$ determined up to unitary isomorphism by its algebraic dimension?
I'm not really sure where to start with this one beyond the observation that it's obviously true for finite-dimensional Hilbert spaces. My intuition would be that the answer is yes, and that a Hilbert space with orthonormal basis of infinite cardinality $\kappa$ has algebraic dimension $2^\kappa$, but I don't actually know how to show that even in the case where $\kappa = \aleph_0$ and $\mathscr{H} = \ell^2(\mathbb{N})$.
