Linear Algebra Done Right · Problem 6A.10

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose

u, v \in V

and

\|u\| \leq 1

and

\|v\| \leq 1

. Prove that

\sqrt{1 - \|u\|^2}\sqrt{1 - \|v\|^2} \leq 1 - |\langle u, v \rangle|.