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?
Best Answer
It really just varies according to the author/instructor. The only universal rule is that we use single bars for absolute values of real (and complex) numbers (e.g.$|-5|$). Once we start defining norms for other objects, we can choose single bars, double bars, or some other notation (although bars are very standard). In some contexts, we use $N(\alpha)$ to indicate a norm of $\alpha$. Reasons for using double bars (or any other notation) might include the desire to differentiate a vector norm, or some other norm, from the absolute value of a scalar.
If you are defining some kind of norm in your own writing, it's a good idea to define your notations before you use them, so that readers can follow your argument, even if they come from a context of using different notations.
(Single bars for absolute value is nearly universal. In some computer systems, however, absolute value of a real number $x$ is denoted $\mathrm{abs}(x)$. There may be other notations floating around, too.)