Veryify the parallelogram law below for vectors u and v in Rn.?

Question

ll u + v ll^2 + ll u - v ll = 2 ll u ll^2 + 2 ll v ll^2

Answer 1

I assume you mean for || u - v || to be squared. If so we have:

ll u + v ll² + ll u - v ll²
= {|| u ||² + 2 || u || || v || + || v ||²} + {|| u ||² - 2 || u || || v || + || v ||²}

= 2 || u ||² + 2 || v ||²