Let $\ell^2(I)= \left\{ x:I\rightarrow \mathbb C\mid \sum _{i\in I} |x_i|^2<\infty \right\} $ with inner product $\sum_{i\in I}x_i \bar y_i$.
I am supposed to find the dimension of this space and for which sets it is separable.
I don't really understand what's going on with the set $I$, but I'm guessing the dimension would just be $|I|$ since the dimension of the usual $\ell^2$ of absolutely summable sequences of complex numbers is of dimension $\aleph _0$. I really don't know how to prove this though - my only intuition is to define a tentative basis $e_i:I\rightarrow \mathbb C$ as just $\delta _i$. For separability, I'm not sure but I think the space can only be separable for $|I|=\aleph _0$.