Denseness of diagonalizable operators
Given this topology, the set of diagonalizable functions is a dense subset of .
We can prove this as follows: Every complex matrix A is conjugate to a matrix B in Jordan canonical form. One can then perturb the diagonal elements bii of B by arbitrarily small numbers εi so that the diagonal elements bii + εiof the perturbed matrix are distinct. But this implies that the perturbed matrix is diagonalizable. Thus, we can find a diagonalizable matrix arbitrarily close to a conjugate of A. But since conjugation is a length-preserving operation on the inner product space of complex matrices, this shows that A is arbitrarily close to a diagonalizable matrix. This completes the proof.