An article I'm reading refers to the $L_2$-norm of a vector $\mathbf{x}$ as $\|\mathbf{x}\|_2$ — the notation is verbatim. The same article refers to $|f'(\centerdot)|$ and $|f''(\centerdot)|$, which resolve to a vector in $\Bbb{R}^p$ and a $p\times p$ matrix, respectively (these are the gradient and hessian of a linear function with $p$ parameters).
I am unclear, though, as to whether the $|\centerdot|$ brackets refer to the $L_1$ or $L_2$-norm, as the notation was never explicitly defined. Do I just assume these are the $L_2$-norm, despite the different notation used earlier for $\|\mathbf{x}\|_2$?