Let's consider an infinite dimensional space $X$ and a linear operator $T$. The resolvent operator of $T$ is $R_\lambda (T) = (T-\lambda I)^{-1}$. A regular value $\lambda$ of $T$ is a complex number such that:
(R1) $R_\lambda(T)$ exists
(R2) $R_\lambda(T)$ is bounded
(R3) $R_\lambda(T)$ defined on a set which is dense in $X$
The resolvent set $\rho(T)$ consists of all regular values $\lambda$ of $T$. The complement $\sigma(T)=C-\rho(T)$ is the spectrum of $T$ and we may distinguish parts of the spectrum:
point spectrum (eigenvalues) $\sigma_p(T)$: (R1) isn't satisfied
continuous spectrum $\sigma_c(T)$: (R2) isn't satisfied, but (R1) and (R3) are satisfied
residual spectrum $\sigma_r(T)$: (R3) isn't satisfied, (R1) is satisfied, (R2) - doesn't matter
Please, help me to clarify a couple of points:
Question 1: The point spectrum consists of eigenvalues and exists in finite dimensional case. 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 $T$). 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 from it.