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Meaning of the continuous spectrum and the residual spectrum So its meaning seems to be the same as in a finite dimensional case (scaling of eigenvectors that roughly represent orientation of the distortion by T T) What is the meaning of the continuous and the residual spectrum? Question 2: Why do we care about dense in the definitions? I have found a related question but didn't get the exact answer
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