# real analysis – Can you check my proof of this equality?

### Prove that

$$lVert x-yrVert^2+lVert x+y rVert^2 = 2 lVert x rVert^2+2 lVert y rVert^2 \ ,text{for all } x,y in mathbb{R}^p$$

#### My proof:

I will manipulate right-hand side of equation to make it equal to the LHS by adding and subtracting $$2 lVert xrVert lVert yrVert$$:
$$2 lVert x rVert^2+2 lVert y rVert^2$$
$$2 lVert x rVert^2+2 lVert y rVert^2 + 2 lVert xrVert lVert yrVert – 2 lVert xrVert lVert yrVert ​$$
$$lVert x rVert^2+ 2 lVert xrVert lVert yrVert + lVert y rVert^2 – 2 lVert xrVert lVert yrVert ​+ lVert x rVert^2 + 2 lVert xrVert lVert yrVert +lVert yrVert^2$$
$$lVert x+y rVert ^2 + lVert x-y rVert ^2$$
Thus, RHS is equal to LHS.

#### The problem

It says that equality should hold for all $$x,y in mathbb{R}^P$$. I am certain that this proof is correct for $$p=1$$ and $$p=2$$, but in higher dimensions I just don’t know. Can anyone highlight the weak side of the proof and give an advice on how to make it more solid.