Linear Algebra Done Right · Problem 7F.11

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms-diagonalizable

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

. Prove that for every

\epsilon > 0

there exists a diagonalizable operator

S \in \mathcal{L}(V)

such that

0 < \|T - S\| < \epsilon