Let u ve Rn u u2 luu2 2u2 2k2 This is called the paralle

Let u, ve Rn. ||u + u|^2 + l|u-u||^2 = 2||u||^2 + 2|k||2. (This is called the parallelogram law). Make a sketch that illustrates this exercise for R. R^2 and R^3.

Solution

1.

We know, ||x||^2=x.x , . represents the dot product.

||u+v||^2+||u-v||^2=(u+v).(u+v)+(u-v).(u-v)=2u.u+2v.v=2||u||^2+2||v||^2