Something has been bugging me lately; maybe you guys can help satisfy my curiosity.
Let $\kappa$ denote any nonzero cardinal. For convenience, we identify $\kappa$ with its smallest ordinal representation. By "a (real) Hilbert space over $\kappa$" we mean a Hilbert space with an orthonormal basis of dimension $\kappa$, which we will represent as the space $$\ell_2(\kappa)=\left\{x\in\mathbb{R}^\kappa:\sum_{\alpha\in\kappa}|x_\alpha|^2<\infty\right\}$$ endowed with the $\ell_2$-norm. Equivalently, $\ell_2(\kappa)$ is the $\ell_2$-closed linear span of unit vectors $(e_\alpha)_{\alpha\in\kappa}$ in $\mathbb{R}^\kappa$.
When $\kappa\leq\mathfrak{c}$, we obtain that $|\ell_2(\kappa)|=\mathfrak{c}$, and when $\kappa=2^\lambda$ for some $\lambda>\aleph_0$ we have $|\ell_2(\kappa)|=\kappa$. However, the general case eludes me, and so I ask the following.
Question 1. What is the cardinality of $\ell_2(\kappa)$?
Since algebraic dimension coincides with cardinality in the remaining cases, we could simply ask for that:
Question 2. What is the algebraic dimension of $\ell_2(\kappa)$ when $\kappa>\mathfrak{c}$?
It is clear that we can identify $\ell_2(\kappa)$ with a subset of $(\mathbb{R}\times\kappa)^\mathbb{N}$, so that $|\ell_2(\kappa)|\leq\kappa^{\aleph_0}$.
This question isn't important or anything, but I was curious about it.
Thanks!
It's just $\kappa^{\aleph_0}$, with more possibly being known based on the value of $\kappa$ and the background set-theoretic assumptions.
It's not hard to show that in fact $\vert l_2(\kappa)\vert=\kappa^{\aleph_0}$; you've already got one half (the upper bound), and for the other half given an $f:\omega\rightarrow\kappa$ let $x_\alpha=2^{-f^{-1}(\alpha)}$ for $\alpha\in ran(f)$ and $x_\alpha=0$ otherwise.
However, even in the presence of GCH this isn't generally simplifiable (contra your sentence beginning "Under GCH ...")! Konig's theorem implies for example that $(\aleph_\omega)^{\aleph_0}>\aleph_\omega$ even if GCH holds. More generally we'll see something interesting happen whenever $\kappa$ has countable cofinality; GCH rules out interesting things happening in all other cases.