Linear Algebra Done Right · Problem 6A.31

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

u, v, w \in V

. Prove that

\left\|w - \tfrac{1}{2}(u + v)\right\|^2 = \frac{\|w - u\|^2 + \|w - v\|^2}{2} - \frac{\|u - v\|^2}{4}.