Linear Algebra Done Right · Problem 7D.1

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

\dim V \geq 2

and

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

. Prove that

S

is an isometry if and only if

Se_1, Se_2

is an orthonormal list in

W

for every orthonormal list

e_1, e_2

of length two in

V