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

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?

• Yes, it's absolute value. – avs Mar 29 at 16:44
• Note that in LaTeX, $\|a\|$ (\| a \|) should be used over $||a||$ (|| a ||) when typesetting vector norms. Compare the readability of $\| v \| \|u \|$ vs. $||u||||v||$. – Brian Mar 29 at 16:51

## 2 Answers

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.

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.