Cauchy Schwarz inequality and absolute value

Question

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$

Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?

Answer 1

Note that $\langle u,v\rangle$ need not be even a real number in general. For example, consider $\mathbb{C^2}$, $u=(1,i)$, $v=(0,1)$.

Even if you are dealing with real inner product spaces, the inner product of $u$ and $v$ need not be positive.

For example, in $\mathbb{R^2}$ take $u=(1,0)$, and $v=(-1,1)$.