When to use $\|x-y\|$ or $|x-y|$ or $d(x,y)$? I am having a hard time understanding when to use $\|x-y\|$ or $|x-y|$ or $d(x,y)$?
I thought that $\|x\|$ was used for matrices or vectors, $|x|$ was used for scalars, and $d(x,y)$ was used for a metric space? Are there other restrictions for what spaces we can use which? ie $\|x\|$ in a normed space... 
I think I am just confusing myself the more I think about it. Does anyone have a good rule of thumb I can use to know when to use a norm, absolute value, or distance?
 A: $\|\cdot\|$ is used primarily for normed spaces.
$d(x,y)$ is used for any metric, so it covers situations like $\|x-y\|$ (for a norm $\|\cdot\|$).
$| \cdot |$ is used for special norms like absolute value (real numbers) and modulus (complex numbers), as well as for things unrelated to norms (e.g., cardinality). Some authors also use $|\cdot|$ for Euclidean norm (e.g., Evans's PDE book) in lieu of $\|\cdot \|$.
I would not spend too much time worrying about "proper" notation, as conventions vary from author to author. Just pick a convention and be consistent within your own paper/work. But it is good to be aware of common usage (as you seem to already be).
A: Actually they are both right. You must understand the notation used at the context to know what you'll use.
The "Analysis on Manifolds" of munkres use $|\cdot|$ to represent the euclidian norm, while other books use $|| \cdot||$ and some books use $|\cdot|_{2}$
The only rule you need to know is to define well what you are writting and be consistent in your notation in the context.
