Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis.
This is Problem 13 in Chapter II in Reed & Simon, and I'm really stuck on this one. Would something like $\{0,1\}^{\mathbb{N}}$ work? I'm not sure this satisfies all the properties, though.
Let $\mu$ be counting measure on $\mathbb{R}$. Then $L^2_{\mu}(\mathbb{R})$ is uncountable with complete orthonormal basis $\{ e_{r} \}_{r\in\mathbb{R}}$, where $e_{r}(x)=0$ for $x \ne r$ and $e_{r}(r)=1$. Every $x \in L^2_{\mu}(\mathbb{R})$ is a function that is non-zero for at most countably many points of $\mathbb{R}$, with norm $\|x\|=\sqrt{\sum_{t\in\mathbb{R}}|x(t)|^2}$.