What's the relationship between $|\cdot|$ and $||\cdot||$ in vector spaces? What's the relationship between $|\cdot|$ and $||\cdot||$ in vector spaces?
I believe using $||\cdot||$ for functions, e.g. $f(x,y) = \frac{xy}{x^2+y^2}$ makes no sense (has no definition), whereas $|\cdot|$ does. But $|\cdot|$ doesn't make sense for vectors, e.g. $(3,4)$,  whereas $||\cdot||$ does.
So what's the relationship?
Specifically, $|\cdot|$ does generalise to $||\cdot||$, but does the converse hold (i.e. do inequalities shown in $||\cdot||$ reduce to $|\cdot|$ somehow)? For $a \in \mathbb{R}$ it makes sense to say $||a||=\sqrt{a^2}=|a|$, what about in $\mathbb{R^n}$?
 A: $|\cdot|$ is usually used for the norm (absolute value) on the base field ($\mathbb{R}$ or $\mathbb{C}$), whereas $\|\cdot\|$ denotes the norm in a vector space. If there are multiple vector spaces $X, Y, Z$ then the norms are sometimes written $\|\cdot\|_X$, $\|\cdot\|_Y$, $\|\cdot\|_Z$.
Why do we keep them separate? One example of where this helps is in the definition of a norm, we have for any vector $v$ and scalar $\alpha$,
$$
\| \alpha v \| = |\alpha| \| v\|
$$
This formula is certainly clearer when you distinguish between the field absolute value and the vector space norm. Otherwise it would look like $\| \alpha v\| = \|\alpha\| \|v\|$.
A: In my experience, pure mathematicians like $ \lVert \cdot \rVert $ more than applied mathematicians and physicists, so


*

*Both would use $\lvert \cdot \rvert$ for real and complex numbers (ordinary absolute value)

*Depending on field, mathematicians might use either for the Euclidean norm $\sqrt{\sum_i x_i^2}$ in $\mathbb{R}^n$, but certainly physicists will use $\lvert \cdot \rvert$ in e.g. $r = \lvert \mathbf{x}\rvert$.

*For anything more general, everyone will use $\lVert \cdot \rVert$, whether it be a more general norm on $\mathbb{R}^n$ or the norm on a different space.


There was one occasion where I saw a lecturer use $\lVert \cdot \rVert$ and $||| \cdot |||$ for two norms, but the less said about that the better: he ended up with something like
$$ \big\lvert \lVert x \rVert - \lvert\lvert \lvert x \rvert \rvert \rvert  \big\rvert, $$
which was just not readable.
