Linear Algebra Done Right · Problem 7F.4

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms

Problem

Suppose

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

v \in V

, and

\|Tv\| = \|T\| \|v\|

. Prove that if

u \in V

and

\langle u, v \rangle = 0

, then

\langle Tu, Tv \rangle = 0