Is anybody aware of, or can provide at least an outline, of a proof that the Hilbert space of Lebesgue functions square-integrable on the closed real interval [a,b], equipped with the $L^2$ norm, is separable?
I've seen an ugly proof involving truncated functions so I'm not desperate, but would really like to use something nice. By the way, if you refer to a particular dense countable subset, could you please explain why it is dense and countable even if you consider it to be a fairly 'high-profile' set?
Thanks