Linear Algebra Done Right · Problem 6C.4

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-complements

Problem

Suppose

e_1, \ldots, e_n

is a list of vectors in

V

with

\|e_k\| = 1

for each

k = 1, \ldots, n

and

\|v\|^2 = |\langle v, e_1 \rangle|^2 + \cdots + |\langle v, e_n \rangle|^2

for all

v \in V

. Prove that

e_1, \ldots, e_n

is an orthonormal basis of

V

. This exercise provides a converse to 6.30(b).