Pythagorean theorem in inner product spaces

- Let $X$ be an inner product space (over $\mathbb{R}$ or $\mathbb{C}$ ) and $x,y\in X$ two orthogonal vectors. Then

\|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}.

Proof : As $x\perp y$ one has $\langle x,y\rangle=0$ . Then

$\displaystyle\\|x+y\\|^{2}$	$\displaystyle=$	$\displaystyle\langle x+y,x+y\rangle$
	$\displaystyle=$	$\displaystyle\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y% ,y\rangle$
	$\displaystyle=$	$\displaystyle\\|x\\|^{2}+\langle x,y\rangle+\overline{\langle x,y\rangle}+\\|y\\|^% {2}$
	$\displaystyle=$	$\displaystyle\\|x\\|^{2}+\\|y\\|^{2}\qquad\qquad\qquad\quad\square$

$Remark-$ This theorem is valid (with the same proof) for spaces with a semi-definite inner product.

Title	Pythagorean theorem in inner product spaces
Canonical name	PythagoreanTheoremInInnerProductSpaces
Date of creation	2013-03-22 17:32:13
Last modified on	2013-03-22 17:32:13
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46C05
Synonym	Pythagoras theorem in inner product spaces
Related topic	PythagorasTheorem