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What's the difference between these two signes $|\overrightarrow{v}|$ and $\|\overrightarrow{v}\|$ for a given vector $\overrightarrow{v}$?

emacs drives me nuts
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Artashes
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    Did you check to see if these were defined in your text? – Cameron Williams Jun 17 '22 at 14:48
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    They both usually mean the length of the vector. Different texts and teachers use the one they prefer. Personally, I think the double lines version is clearer, but know I might have to deal with the single line version. – JonathanZ Jun 17 '22 at 14:49
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    Is $A$ a vector, or is it a matrix? The "usual" notation is $A$ for a matrix and $v$ for a vector of a vector space $V$. For a matrix, $||A||$ is a matrix norm. – Dietrich Burde Jun 17 '22 at 14:50
  • Good point, @DietrichBurde. I assumed it was certain we were dealing with a vector, but I'm also used to capital letters being matrices too. With double checking. – JonathanZ Jun 17 '22 at 14:53
  • do we have to define these signes before using them?? I mean you don't define "=" as equality sign because it is obvious. I thought it's the same for |...| and ||...|| – Artashes Jun 17 '22 at 14:54
  • @DietrichBurde, just modified my post, to make it clear that $\overrightarrow{v}$ is a vector – Artashes Jun 17 '22 at 14:57
  • Yes, I think we need to define these "signs". Just now the following post appeared here. It has the notions $|u|$ and $||\nabla u ||_2$ and so on. I am not sure what exactly is meant without a definition. – Dietrich Burde Jun 17 '22 at 15:09
  • @DietrichBurde, the use of each sign is a covenetional I think, $|...|$ is used for absolute value, or modulus of a complex number, while $||...||$ is used for normes. – Artashes Jun 17 '22 at 15:31
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    Those conventions are not as strict as you seem to think, which you will discover as your mathematical reading broadens. Keep in mind, the absolute value IS an example of a norm, and its "single bar" notation $|...|$ is often applied to more general norms. – Lee Mosher Jun 17 '22 at 15:41

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If $\overrightarrow v$ is an element of $\mathbb R^n$ for some positive integer $n$, then these two notations are the same: choice of $\|\cdot\|$ or $|\cdot|$ depends on the author.

I specified the meaning of $\overrightarrow v$ above because the word "vector" has many different uses in mathematics.

Examples. If I am talking about $\mathbb R^3$, I may want to use $|\lambda|$ for the absolute value of a scalar and $\|\overrightarrow v\|$ for the norm of a vector.

If I am talking about Hilbert space $L^2(\Omega,\mathbb R^n)$, then I may want ot use $|\overrightarrow v|$ for the norm of $\overrightarrow{v} \in \mathbb R^n$ and $\|\varphi\|$ for the norm of $\varphi \in L^2(\Omega,\mathbb R^n)$. Like this. $$ \|\varphi\| := \left(\int_\Omega|\varphi(t)|^2\;dt\right)^{1/2} $$

GEdgar
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