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What is the mathematical terminology if I want to express the number of zeros (or non-zero) elements in a vector?

There is a $l_0$ "norm" that counts non-zero elements of a vector in $\mathbb{R}^n$ $$\sum_{i=1}^{n}|x_i|^0$$ however it is a problematic function because we have to define $0^0=0$. Also it is not a norm but just called in quotes "norm".

Shall we use text to describe a zero-counting function $f(x)$? We could define without mathematical operators:

$f(x)$ is the function that counts the number of $0$ in $x$

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    $\begingroup$ Also, what you call the $\ell^0$ norm is not, in fact, a norm. $\endgroup$
    – Thomas Andrews
    Commented Oct 18, 2023 at 19:34
  • $\begingroup$ There probably isn't a good name for this, because a vector is not, in general, a sequence of values. That is just the most common way of representing vectors, given a known basis. $\endgroup$
    – Thomas Andrews
    Commented Oct 18, 2023 at 19:35
  • $\begingroup$ I think that correcting codes use this kind of "norm" , you should check that $\endgroup$
    – julio_es_sui_glace
    Commented Oct 18, 2023 at 19:37
  • $\begingroup$ For a vector that's just 0s and 1s there are a bunch of possibilities here: math.stackexchange.com/questions/4661812 $\endgroup$
    – JonathanZ
    Commented Oct 18, 2023 at 19:42
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    $\begingroup$ To avoid $0^0$ exactly, you can take limits: $$\lim_{p\to0^+}\sum_{i=1}^n|x_i|^p,$$ noting that $0^p=0$ for $p>0$ (but not for $p<0$). $\endgroup$
    – mr_e_man
    Commented Oct 18, 2023 at 21:14

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