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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?

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

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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.

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