Is there any difference between the absolute values operators $|z|$ and $\|z\|$?

Question

Is there any difference between the absolute values operators $|z|$ and $\|z\|$ where $z=a+ib$?

Answer 1

Usually, no. But if you see both notation used in the same discussion, it is possible that the author intended to define $\|z\|$ to be a different norm; in that case it will strictly depend on context.

Answer 2

Usually, $\|z\|$ is defined on vectors, while $|z|$ is defined on scalars. Nevertheless, since $z$ is also a one-dimensional vector (over $\mathbb C$) or a two-dimensional real vector, we have $|z| = \|z\|$.

Answer 3

I have only seen $|\cdot |$ used in the context of complex numbers. If you think of a complex number as a real planar vector, the planar vector's $\|\cdot \|$ is the same as the corresponding complex number's $|\cdot |$.