Meaning of the continuous spectrum and the residual spectrum @Konstantin : The continuous spectrum requires that you have an inverse that is unbounded If X X is a complete space, then the inverse cannot be defined on the full space It is standard to require the inverse to be defined on a dense subspace If it is defined on a non-dense subspace, that falls into the miscellaneous category of residual
is bounded linear operator necessarily continuous? 3 This property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator Yes, a linear operator (between normed spaces) is bounded if and only if it is continuous