Question

Proving the Cauchy Schwartz inequality.

Answer 1

Hi, I was just wondering why my book is proving it like this (see photo).

Why isn't it sufficient to do it like this instead? \[|\mathbf{x}\cdot\mathbf{y}|\leq|\mathbf{x}||\mathbf{y}|\]\[\frac{|\mathbf{x}\cdot\mathbf{y}|}{|\mathbf{x}||\mathbf{y}|}\leq1\]\[\frac{|\mathbf{x}||\mathbf{y}|}{|\mathbf{x}||\mathbf{y}|}\cos\theta\leq 1\]\[\cos\theta\leq 1\]which is true.

Answer 2

I thought \[\large \bf \left| x.y \right| =\left| x \right| \left| y \right|\]

always .Why do you start with \(\bf \large \le \) here ?

Answer 3

In general, whenever you start a proof, you never want to start with the statement you are trying to prove. So starting with:\[|x\cdot y|\le \|x\|\|y\|\]would be considered incorrect. Also, that book seems to not be using the fact that:\[x\cdot y=\|x\|\|y\|\cos \theta\], so that is why they use this other method. The reason for not using that formula is that it only makes sense if you are dealing with a space that has a geometric representation. If you are dealing with a space of functions, what does it mean for two functions to be separated by an angle of theta? The method of proof presented in the book works for any vector space, with any inner product.

Answer 4

thanks joemath!

and AravinG. It's only an equality if the vectors are linearly dependent I think.