So bound states have continuous position representation, yet countable discrete energy levels. Doesn't this imply that one of the following operations is invalid: <x|> or <E|>. Because |> cannot simultaneously have countable and uncountable number of elements. And the number of elements in the bra and the ket need to match otherwise the operation doesn't make sense.
The ket |> exists independently of whatever representation we choose. So what's going on here?
