Linear Algebra Done Right · Problem 6A.14

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

v \in V

and

v \neq 0

. Prove that

v/\|v\|

is the unique closest element on the unit sphere of

V

v

. More precisely, prove that if

u \in V

and

\|u\| = 1

, then

\left\|v - \frac{v}{\|v\|}\right\| \leq \|v - u\|,

with equality only if

u = v/\|v\|