What do double bars around a vectors name indicate? For example they are used at the Triangle inequality page of wikipedia. I am struggling to understand a proof written elsewhere.
If $\displaystyle \alpha$ is a scalar and $\displaystyle \vec{v}$ vector then $\displaystyle
\left\| {\alpha \vec{v}} \right\| = \left| \alpha \right|\left\| \vec{v} \right\|$
Is that what you mean by using ‘both’ notations on the same page?
If so those are not the same. One is the absolute value of a scalar(a number) the other is length of a vector.
As was said, some authors use only the single bar for both concepts.
Also one possibility, which I doubt is the case, but is entirely possible, consider a vector of one dimension $\displaystyle V = <-2>$ and the scalar $\displaystyle -2$, the following is true:
$\displaystyle ||V||=|-2|$. This is so because $\displaystyle ||V||=\sqrt{(-2)^2} = \sqrt{4} = 2$ and $\displaystyle |-2| = 2$. So, in a strange abuse of notation, the absolute value of the componant of a one dimensional vector is equal to its magnitude. Now, I doubt that wikipedia would use such a rare case, and such an "abusive of notation" to denote this fact. I'm sure its highly more likely that its simply the alternative notation for the magnitude of a vector. But its interesting to see a relationship between the two notations based on the fact that:
$\displaystyle |a| = \sqrt{a^2}$
Thats my two cents, take it for what its worth.