What's the difference between $|\overrightarrow{v}|$ and $\|\overrightarrow{v}\|$

Question

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

Answer 1

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.

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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. $$ \|\varphi\| := \left(\int_\Omega|\varphi(t)|^2\;dt\right)^{1/2} $$