For any x,y in R^2 show that, IIx + yII^2 + IIx - yII^2 = 2(IIxII^2) + 2(IIyII^2)?

Question

Also show that, IIx + yII = IIx - yII iff IIx + yII^2 = IIxII^2 + IIyII^2.

I've been able to prove the first part i.e. IIx + yII^2 + IIx - yII^2 = 2(IIxII^2) + 2(IIyII^2) but I'm confused about the second part. The first part I was able to prove using IIx + yII^2 = IIxII^2 + IIyII^2, how do I prove IIx + yII = IIx - yII?

Answer 1

||x + y||^2 = ||x||^2 + 2 + ||y||^2
||x - y||^2 = ||x||^2 - 2 + ||y||^2

Adding the two equations you can see that:
IIx + yII^2 + IIx - yII^2 = 2(IIxII^2) + 2(IIyII^2)

Subtracting one from another:
||x + y||^2 - ||x - y||^2 = 4

Now, both norms are equal iff =0
iff ||x + y||^2 = ||x||^2 + 2 + ||y||^2 = ||x||^2 + ||y||^2

Answer 2

IIx + yII=||x|+|y||

IIx - yII=||x|+|-y||=|x|+|y||=IIx + yII (Q.E.D.)