1. Inner product space

How ca I prove that in a real/complex vector space V
$\displaystyle ||x + y||^{2}+ ||x - y||^{2} = 2(||x||^{2} + ||y||^2)$
Also If I consider the parallelogram
with adjacent sides OP; OQ where P is the point (x1; x2); Q is the point (y1; y2) and O is
the origin.What does this say about a parallelogram in the plane?
This is an exercisse from LINEAR ALGEBRA AND ITS APPLICATIONS David C. Lay.

2. Originally Posted by StefanM
How ca I prove that in a real/complex vector space V
$\displaystyle ||x + y||^{2}+ ||x - y||^{2} = 2(||x||^{2} + ||y||^2)$
Just consider this: $\displaystyle \|x+y\|^2=(x+y)\cdot(x+y)$.
Now expand to get $\displaystyle x\cdot x+2x\cdot y+y\cdot y.$
Do that for $\displaystyle \|x-y\|^2$ and add.

3. I was confuse about the fact that $\displaystyle ||x^{2}||=<x,x>$...How can I solve the parallelogram problem?

4. Originally Posted by StefanM
I was confuse about the fact that $\displaystyle ||x^{2}||=<x,x>$...How can I solve the parallelogram problem?
You asked about proving $\displaystyle \|x+y\|^2+\|x-y\|^2=\|x\|^2+2\|x\|\|y\|+\|y\|^2$.
It is a fact that $\displaystyle \|x+y\|^2=<x+y,x+y>$, which can be expanded.
$\displaystyle \|x+y\|~\&~\|x-y\|$ are the lengths of the diagonals of a parallelogram in the plane.