Generalized eigenspaces for distinct eigenvalues form a direct sum

For a linear operator $f$, the generalized eigenspacesbelonging to distinct eigenvalues are linearly independent. On the generalized $\alpha$-eigenspace, $f-\beta I=(\alpha -\beta )I+N$ is invertible for $\beta \ne \alpha$ because the nilpotent part $N$ has a finite geometric-series inverse.