Let $f:\mathbb{R}^N\to\mathbb{R}$ be a $\mathcal{C}^1$ function. To mean "the modulus of the gradient of $f(x)$" I usually use the notation
$$|\nabla f(x)|.$$
Reading a certain proof I see the notation $|\nabla f|(x)$, which I don't know what it means. It is the same thing as $|\nabla f(x)|$ or does it mean something else?
I hope someone could help.
Thank you in advance!
Best Answer
Let me answer since I do not (did not) have enough reputation to turn it into a comment. Yes, it is usually the same. The second notation is just to highlight that it can be considered as a function itself, like $g:= |\nabla f|$, depending on $x$.