I have come across the following example of a non-separable Hilbert space:
Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ defined on the $\sigma$-algebra $\Bbb P(I)$ of all subsets of $I$. Then $\ell^2(I)=L^2\big(I,\Bbb P(I),\lambda_{\text{ count}}\big)$ is a Hilbert space, and it comprises all functions $a:I\to\Bbb R$ (or $\Bbb C$) for which the support $F=\{i\in I : a(i)\ne0\},$ is finite or countable, and for which $\sum_{i\in I}|a_i|^2=\sum_{i\in F}|a_i|^2\lt\infty$.
Why do I need the discrete topology on $I$? Or more generally: why do I need a topology? If we talk about $L^p$ spaces in general, we only want a measure space and we don't mention a topology because $f \in L^p$ doesn't have to be continuous. Thanks for your help.