Uncountable infinite dimensional Hilbert space

[eztum]

When I had to deal with non-seperable Hilbert spaces, decades ago, all interested students
knew the (in a sense trivial) example cited here by dextercioby and the non-trivial example of 'almost-periodic functions'. Also everybody knew that the Fock space over a seperable Hilbert space (acting as 'one-particle space') is separable. Today I 'know' that all Hilbert spaces are finite-dimensional! Tempora mutantur!