I was thinking about this proof of the cauchy-schwarz inequality,
I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$.
We know that,
$$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ is the angle between $u$ and $v$.
But $-1 \leq \cos{\theta} \leq 1$, so, $$0\leq|\langle u,v\rangle|\leq|u||v|$$ where the equality holds if and only if $|\cos{\theta}|=1$, that is, $u$ and $v$ are linearly dependent.
This is correct? For which spaces? Only for $\mathbb{R^n}$?
Thanks :]