Linear Algebra Done Right · Problem 7A.21

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Suppose

D \colon \mathcal{P}_8(\mathbf{R}) \to \mathcal{P}_8(\mathbf{R})

is the differentiation operator defined by

Dp = p'

. Prove that there does not exist an inner product on

\mathcal{P}_8(\mathbf{R})

that makes

D

a normal operator.