1. ## Inner product space

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

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

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