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Which symbol is more commonly used to denote the Euclidean norm: $ \left \| \textbf a \right \| $ or $ \left | \textbf b \right |$?

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    I don't know which one is more common, but the double vertical lines are sometimes preferred because the single lines usually are used with the abs. value / norm of the real or complex numbers2012-08-23
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    @Thomas And I was wondering....isn't Euclidean distance essentially the same as absolute value anyway? I mean, if we're talking about the maximum norm, then I can understand the avoidance of the single-vertical-line notation...2012-08-23
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    Yes, there are similarities. Let me write an answer for this.2012-08-23
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    Somehow, whenever I see $|\cdot|_2$, it often has an $n$ on its subscript.2012-08-24

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As mentioned above, I don't know what is most common (statistically). However, ff you have a vector $V$ space over say the real numbers $\mathbb{R}$, then you can have a norm $\|\cdot\|$ on the vector field (so you get a normed space). One thing that you would like is: $$ \| \alpha v\| = \lvert\alpha\lvert\cdot\| v\|. $$ for $\alpha \in \mathbb{R}$, and $v\in V$. Here the single vertical lines is the norm on the real numbers and the double lines is the norm on the vector space.

If you consider for example the real numbers as a vector space over itself, then you can use the absolute value as a norm.

If you have the vector space $V = \mathbb{R}^n$ as a vector space over the real numbers, then I do believe that the standard notation is the doube lines $\|\cdot \|$. Again, this is because you want to have the single lines for the real numbers. Note that even though the absolute value and the norm seem like the same thing, they are different because the absolute value is evaluated at real numbers, the norm of the vectors. Indeed the Euclidean norm is defined from the absolute value. So for $v = (v_1, \dots, v_n)$, you have $$ \| v \|^2 = \lvert v_1\lvert^2 + \dots + \lvert v_n\lvert^2. $$ Note what happens when $n = 1$, then you have $\|\cdot \| = \lvert\cdot\lvert$.

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    Hey thanks. I was referring specifically to Euclidean space, i.e. $\mathbb{R}^n$. In this case, isn't Euclidean norm essentially absolute value? But for a general vector space, I will use the double-vertical symbol. I just haven't decided which symbol to adopt for Euclidean distance.2012-08-23
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    $Ryan, ahh I see. Let me update the answer then.2012-08-23
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    Thanks Thomas. But it does seem that the confusion in $\mathbb R^n$ doesn't arise if vectors are appropriately distinguished from scalars. Ironicallly, using the double vertical still seems to be more potentially confusing because it could refer to any type of norm and not necessarily distance.2012-08-23
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$ \| \mathbf{v} - \mathbf{u} \| $

or

$ \| \mathbf{v} - \mathbf{u} \|_2 $

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    You might want to add some more to this answer to make it more clear what you are saying.2012-08-23