Understanding Schwarz Inequality and Its Role in Higher Dimensions

[ahmed markhoos]

Hello,

I'm having a small problem with Schwarz inequality, |u⋅v|≤||u||||v||

the statement is true if and only if cosΘ≤1 !, I'm familiar with this result but how could it be more than 1?
what is so special in higher dimensions that it gave the ability for cosine to be more than 1? why and how?

 

[Fredrik]

It can't be more than 1. The inequality $|\langle u,v\rangle|\leq\|u\|\|v\|$ holds for all vectors u and v. That's why it makes sense to define "the angle between u and v" as the $\theta$ such that
$$\cos\theta =\frac{\langle u,v\rangle}{\|u\|\|v\|}.$$ This isn't some new version of the cosine function. It's the plain old cosine function that you're familiar with.