Linear Algebra Done Right · Problem 7F.21

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms-inner-product

Problem

Suppose

\dim V > 1

and

\dim W > 1

. Prove that the norm on

\mathcal{L}(V, W)

does not come from an inner product. In other words, prove that there does not exist an inner product on

\mathcal{L}(V, W)

such that

\max\{\|Tv\| : v \in V \text{ and } \|v\| \leq 1\} = \sqrt{\langle T, T \rangle}

for all

T \in \mathcal{L}(V, W)