Here's an equation from a text book for computing a unit vector: $$\hat v = \frac{\overline v}{ \sqrt{ \sum ^n _{i=1} (\overline v_i)^2 } }$$
Now I may be wrong here, but using $n$ doesn't really cut it here for me. $n$ can't be any old amount, it has to be the number of dimensions in $ \overline v$, right? So is there a symbol for the number of dimensions in a vector?
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$\begingroup$Implicit in the question is that the vectors are in $\mathbb{C}^n$, which is a particular type of vector space. There are many others, e.g. various types of function space, polynomial space, matrix space. None of these have "coordinates" as such in their representations -- this exists only in coordinate spaces such as $\mathbb{R}^n$ or $\mathbb{C}^n$. However to have a coordinate space you must specify how many coordinates there are; hence the notation requested is superfluous.
Note: obviously some authors omit specifying what the ambient space is such as apparently the author of the text in question; however using the letter $n$ for the dimension of the coordinate space is so common as to be considered standard.
$\endgroup$ $\begingroup$Notation typically identifies a space's dimension, not a vector's space's dimension; we write $\dim V$, not $\dim\operatorname{Space}v$ or anything like that. If $n$'s inclusion here bothers you, I recommend rewriting the sum, e.g. as $v^\ast\cdot v$ or $\sum_i\overline{v}_i^2$. (Without explicit constraints on $i$, $\sum_i$ means "sum over all relevant $i$", for a suitable definition of relevance that's obvious in context).
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