What is an example of a Hilbert Space that is not any subset of ${\mathbb R}^n$, $\mathbb{C}^n$ or $L^2$ (n-dimensional reals, n-dimensional complex numbers, or Lebesgue integrable functions)?
I'm looking for an example that is different from the usual spaces we learn about.
On
Sobolev Spaces
https://en.m.wikipedia.org/wiki/Sobolev_space
One of simplest ones the set of real valued differentiable functions on [0,1] where the norm $||x||_2 + ||x’||_2$ is finite.
Maybe a Hilbert space with a non-countable basis: for example, given a non-countable set $\Gamma$, take $$\ell_2(\Gamma) =\{x:\Gamma\to{\mathbb C}\text{ or }{\mathbb R}:\, \sum_{\gamma\in\Gamma}|x(\gamma)|^2<\infty\},$$ where $\sum_{\gamma\in\Gamma}$ is defined as the supremum over all possible finite sums.
Recall that any two Hilbert spaces are isomorphic iff its basis have the same cardinality.