Why use absolute value for Cauchy Schwarz Inequality? I see the Cauchy-Schwarz Inequality written as follows 
$$|\langle u,v\rangle| \leq \lVert u\rVert \cdot\lVert v\rVert.$$
Why the is the absolute value of $\langle u,v\rangle$ specified? Surely it is apparent if the right hand side is greater than or equal to, for example, $5$, then it will be greater than or equal to $-5$?

What if the inner product is a complex number which can happen if $u$ and $v$ are vectors of
complex numbers?
For real vectors, the Cauchy-Schwarz Inequality is better written as
$$-||u||\cdot ||v|| \leq \langle u, v \rangle \leq ||u||\cdot ||v||$$
where, if $||v|| > 0$, then equality holds in the right inequality
if $u = \lambda v$ with $\lambda \geq 0$
and in the left inequality if  $u = \lambda v$ with $\lambda \leq 0$.
