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In the article on $\ell_p$-space, Wikipedia gives two alternatives for the $p\to 0$ limit of a $p$-norm. One is

$$\lVert x\rVert_0 = \sum_n 2^{-n}\frac{\vert x_n\rvert}{1+\lvert x_n\rvert}$$

and the other is the number of nonzero components

$$\lVert x\rVert_0 = \sum_n \lvert x_n\rvert^0$$

with the convention that $0^0=0.$ I suppose this latter definition only works in finite dimensions.

Neither of these definitions makes much sense to me. It seems to me that the definition of the $p$-norm for $p=0$ should be

$$\lVert x \rVert_0 = \prod_n \lvert x_n\rvert.$$

The exact same calculation that shows that the $p\to0$ limit of the Hölder mean is the geometric mean seems to work here. Just as we show by passing through an exponentiation that

$$\lim_{p\to 0} \left(\frac{1}{n}\sum_n\lvert x_n\rvert^p\right)^{1/p} = \left(\prod_n |x_n|\right)^{1/n},$$

an almost identical computation seems to show that

$$\lim_{p\to 0} \left(\sum_n\lvert x_n\rvert^p\right)^{1/p} = \prod_n |x_n| = \exp\left(\sum_n \log \lvert x_n\rvert\right).$$

Wouldn't it make the most sense to take this as the definition of the $0$-"norm" (quotes here because the $\ell_p$ norm is only a norm for $1\leq p \leq \infty$), analogous to the $0$-mean/geometric mean? Or at least offer it as an alternative?

ziggurism
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  • I didn't find any posts answering this before I posted it with stackexchange search, but I did find some using google afterwards. Including https://math.stackexchange.com/questions/642939/what-does-the-lp-norm-tend-to-as-p-to-0, and https://math.stackexchange.com/questions/530639/what-is-the-limit-of-l-p-at-p-0 and https://math.stackexchange.com/questions/115428/limit-of-lp-norm-when-p-to0 and https://math.stackexchange.com/questions/1534870/limit-of-l-p-norm-as-p-rightarrow-0 and https://math.stackexchange.com/questions/1683399/why-define-norm-in-l-p-in-that-way – ziggurism Dec 10 '24 at 01:25
  • On closer inspection, the exact same calculation does not work. For the $p$-mean you have a l'Hopital limit because $\log\left(\frac{1}{n}\sum \lvert x_i\rvert^p\right)$ goes to zero as $p\to 0$. But without the $\frac{1}{n}$ in the corresponding $p$-norm calculation, I guess the numerator goes to $\log n$ instead of $0$, so no l'Hopital. – ziggurism Dec 10 '24 at 01:45
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    Your product formula does not lead to a metric as the triangle inequality fails, unlike the ones from Wikipedia l. – Ryszard Szwarc Dec 10 '24 at 03:10
  • @RyszardSzwarc as I said in the question, the $\ell^p$ "norm" is only a norm for $1 \leq p \leq \infty$. The triangle inequality already fails for $0 < p < 1$. One property that the product retains which the other proposed $0$-norms do not is homogeneity. But there is no expectation of it satisfying the triangle inequality. – ziggurism Dec 10 '24 at 08:34
  • I do not see homogeneity satisfied in your product definition. – Ryszard Szwarc Dec 10 '24 at 09:13
  • First expression is just a metric on the space of all sequences. It follows a standard idea to produce a metric on some space. The second expression is the limit $p\to0$ of $\sum_n |x_n|^p$ (without exponent $1/p$). – daw Dec 10 '24 at 09:59
  • @RyszardSzwarc $\prod (\lambda x_1,\dotsc , \lambda x_n) = \lambda^n \prod (x_1,\dotsc , x_n).$ it is homogeneous of degree $n$. Not homogeneous of degree 1, as a norm would be. – ziggurism Dec 10 '24 at 12:08
  • You have to restrict to finite dimensional space. Moreover the norm vanishes if one of the coordinates vanish. I do not see any advantage of your formula. – Ryszard Szwarc Dec 10 '24 at 12:32
  • @RyszardSzwarc there are convergence criteria which allow infinite products to be defined. but any of the $p$-norms can diverge for a sequence, and that's also true for one of the wikipedia alternative $0$-norms, so it's not an issue. My interest in this formula is because I wanted to find a unified $p$-sum analogous to the unified $p$-mean for all $p$ and from there define a $p$-derivative and integral, which reduces to classical derivative and integral for $p=1$, and multiplicative integral and derivative for $p=0$, and for $p=-1$ give me a novel formula for harmonic derivative and integral – ziggurism Dec 10 '24 at 14:50

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