Linear Algebra Done Right · Problem 7D.14

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

v \in V

with

\|v\| = 1

and

b \in \mathbf{F}

. Also suppose

\dim V \geq 2

. Prove that there exists a unitary operator

S \in \mathcal{L}(V)

such that

\langle Sv, v \rangle = b

if and only if

|b| \leq 1