Do the $|$ around $|\langle u,v\rangle|$ refer to absolute value in the inner product version of the Cauchy-Schwarz inequality?

Question

The full inequality is: $|\langle u,v\rangle| \leq ||u|| ||v||$

I understand that $||$ around the vectors $u$ and $v$ signifies the taking of their norm, but what do the single | around $\langle u,v\rangle$ mean?

Answer 1

Yes, it is absolute value. Note that $\langle u, v \rangle$ is a scalar. In a real vector space, this is a real number, and you are taking its absolute value in the usual way. In a complex vector space, it's a complex number, and you are taking its complex modulus.

Answer 2

Yes, they refer to the absolute value. Since $u$ and $v$ are elements of an inner product space, they require a specific norm. However, $\langle u,v \rangle$ is either a real or complex number, so we use the Euclidean norm.