Hilbert space and uncountable cardinal

Question

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?

Answer 1

Let $\kappa$ be any cardinal. Define $\ell^2(\kappa)$ as follows: As a set, we have $$ \ell^2(\kappa) := \left\{x \colon \kappa \to \mathbf K \Biggm| \sum_{i \in \kappa} |x(i)|^2 < \infty \right\} $$ With the scalar product $$ (x,y) := \sum_{i \in \kappa} x(i)\overline{y(i)} $$ $\ell^2(\kappa)$ is a Hilbert space, of which the maps $e^i$, $i \in \kappa$, given by $e^i(j) = \delta_{ij}$, $j \in \kappa$ form an orthonormal basis of cardinality $\kappa$.