Linear Algebra Done Right · Problem 7D.11

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

S \in \mathcal{L}(V)

. Prove that

S

is a unitary operator if and only if

\{Sv : v \in V \text{ and } \|v\| \leq 1\} = \{v \in V : \|v\| \leq 1\}.