Linear Algebra Done Right · Problem 6A.27

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

V

is a complex inner product space. Prove that

\langle u, v \rangle = \frac{\|u + v\|^2 - \|u - v\|^2 + \|u + iv\|^2 i - \|u - iv\|^2 i}{4}

for all

u, v \in V