TITLE: Is there a difference between one or two lines depicting the norm?
QUESTION [22 upvotes]: I know that the  |  stands for the norm, however I see it sometimes with two lines, sometimes with one. Is there a difference? Do I need to take care when writing a formula?
So basically, is there a difference between

$|a|$ and $\|a\|$ ?

Is it the same when it comes to vectors instead of just numbers?

REPLY [3 votes]: If I posted this as a question it would be a duplicate, so I decided to post it as an answer in the hope that it either gets confirmed or rejected - if the former, it will stay as a valid answer; if the later, it may stay as a counter-example, or be deleted.
Even though it seems as harping on $\vert \cdot \vert$ versus $\Vert \cdot \Vert$ is as pedantic as useless, there is some sense in which it reinforces the concept, and that is in the difference in the space of continuous functions between the absolute value of the function $\vert \cdot \vert,$ as in, for example for $f(x)=\sin x:$

$f(x) = \vert \sin x \vert$

and different norms, e.g.
$$\Vert f(x) \Vert_1 = \int_0^{2\pi}\vert \sin x \vert dx =4$$

vs
$$\Vert f(x) \Vert_2 = \left( \int_0^{2\pi}\vert \sin x \vert^2 dx \right )^{1/2} =\sqrt \pi$$
or
$$\Vert f(x) \Vert_\infty = \underset{0\leq x \leq 2\pi}{\max}\left(\vert \sin x \vert \right ) =1$$