Abstract
“It approaches” is not a property of a sequence alone. This paper sets the three rulers $\ell^1,\ell^2,\ell^\infty$ down in numbers and writes both that they are equivalent in finite dimension and that what the equivalence guarantees decays with the dimension. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, the best constants between the three norms are $\sqrt{n}$ and $n$. At $n=10^6$ the $\ell^1$ and $\ell^\infty$ norms may differ by a factor of a million — and they are still “equivalent” (Sections 2 and 3).
Second, that extreme lies in no ordinary direction. Among two hundred thousand random vectors the largest $\|x\|_1/\|x\|_2$ at $n=1000$ was only $26.21$ against the bound of $31.62$, which is reached only where every coordinate is equal (Section 2).
Third, this is the core. For $x_n=(1/n,\dots,1/n)$ the norm $\|x\|_1$ stays at $1$ and never moves, while $\|x\|_2=1/\sqrt n$ and $\|x\|_\infty=1/n$ both go to zero — one sequence that approaches under one ruler and does not stir under another (Section 4).
Fourth, with functions it is fiercer. A spike of height $n^{1/4}$ and width $1/n$ climbs to $100$ in the sup norm while falling to $0.01$ in $L^2$ and $10^{-6}$ in $L^1$ — one sequence going both to zero and to infinity (Section 5).
Fifth, at $n=50$ the volume of the $\ell^1$ unit ball is $\mathbf{3.04\times10^{64}}$ times smaller than that of the $\ell^\infty$ ball (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the equivalence of norms in finite dimension, Riesz's lemma and the volumes of the $\ell^p$ unit balls are all standard. The equivalence theorem is not proved — this paper measures the constants and cites the theorem. General Banach spaces are not treated — only $\ell^p$ and $C[0,1]$ appear. Neither completeness nor duality is treated. No norm is called the right one — what is written is only what changes when one is chosen. Relation to Paper 177: 177 treated the order of the quantifiers and wrote the fence of compactness — the equivalence here is likewise a question of whether the constant comes before the dimension or after, the same fence from the other side. Relation to Paper 197: 197 treated the four distinct roots of “converges” — what changes here is not the kind of convergence but the way distance is measured. The corpus carries “unit ball” in twenty-five papers, and every one of them is a geometric ball; the unit ball of a normed space appears here for the first time. What is added is setting the best constants side by side at six dimensions, measuring that random samples never approach that extreme, writing out one sequence to which two norms give different answers, watching three norms split into zero, zero and infinity on a sequence of spikes, checking that against integration on a grid, and giving the ratio of unit-ball volumes at four dimensions.
1. Notation and Position
For $x\in\mathbb{R}^n$ put
$$ \|x\|_1=\sum_i|x_i|,\quad \|x\|_2=\Bigl(\sum_i x_i^2\Bigr)^{1/2},\quad \|x\|_\infty=\max_i|x_i|. $$Two norms are equivalent when $a\|x\|_A\le\|x\|_B\le b\|x\|_A$ holds for some positive $a,b$ and every $x$. In finite dimension any two norms are equivalent.
Position: Paper 182 treated the curse of dimensionality as not one curse — what is cursed here is the constant of the equivalence. In Paper 208 curvature did the work of dimension — here dimension alone does the work.
2. Three Norms and the Best Constants
Check 2.1: the inequalities $\|x\|_\infty\le\|x\|_2\le\|x\|_1$ always hold; going back costs the following.
| $n$ | Bound on $\|x\|_1/\|x\|_2$ | Bound on $\|x\|_2/\|x\|_\infty$ | Bound on $\|x\|_1/\|x\|_\infty$ |
|---|---|---|---|
| $2$ | $1.414213562$ | $1.414213562$ | $2$ |
| $10$ | $3.16227766$ | $3.16227766$ | $10$ |
| $100$ | $10.0$ | $10.0$ | $100$ |
| $1000$ | $31.6227766$ | $31.6227766$ | $1000$ |
| $10^6$ | $\mathbf{1000.0}$ | $\mathbf{1000.0}$ | $\mathbf{10^6}$ |
Check 2.2: how rare that extreme is. Two hundred thousand Gaussian vectors at each dimension, taking the largest $\|x\|_1/\|x\|_2$.
| $n$ | Largest among $200000$ | $\sqrt n$ | The all-equal direction |
|---|---|---|---|
| $2$ | $1.41421356$ | $1.41421356$ | $1.41421356$ |
| $10$ | $3.12127801$ | $3.16227766$ | $3.16227766$ |
| $100$ | $8.79139938$ | $10.0$ | $10.0$ |
| $1000$ | $\mathbf{26.20659420}$ | $\mathbf{31.62277660}$ | $\mathbf{31.62277660}$ |
Observation 2.3 A random direction does not even approach the bound. What reaches it is the single direction with every coordinate equal — one point out of the whole sphere. The worst case and the typical case are distinct roots.
3. Equivalent, and the Constant Runs Away
Observation 3.1 $\|x\|_\infty\le\|x\|_1\le n\|x\|_\infty$ is true at every $n$, so $\ell^1$ and $\ell^\infty$ are always equivalent in finite dimension.
Check 3.2: what that guarantee says.
| $n$ | Upper $\div$ lower | Digits they may disagree by |
|---|---|---|
| $1$ | $1$ | $0.00$ |
| $10$ | $10$ | $1.00$ |
| $10^4$ | $10^4$ | $4.00$ |
| $10^6$ | $\mathbf{10^6}$ | $\mathbf{6.00}$ |
Observation 3.3 The theorem is true at each $n$, and says less the larger $n$ is. At a million dimensions it says only that the two rulers may disagree by a factor of a million — and the word “equivalent” remains true. A word being true and a word being useful are two things.
4. The Core: One Sequence That Approaches under One Ruler and Not the Other
Check 4.1: take $x_n=(1/n,1/n,\dots,1/n)$ with $n$ coordinates.
| $n$ | $\|x\|_1$ | $\|x\|_2$ | $\|x\|_\infty$ |
|---|---|---|---|
| $1$ | $1.0$ | $1.0$ | $1.0$ |
| $10$ | $\mathbf{1.0}$ | $0.31622777$ | $0.1$ |
| $100$ | $\mathbf{1.0}$ | $0.1$ | $0.01$ |
| $10^4$ | $\mathbf{1.0}$ | $0.01$ | $0.0001$ |
| $10^6$ | $\mathbf{1.0}$ | $\mathbf{0.001}$ | $\mathbf{10^{-6}}$ |
Observation 4.2 It is one sequence. Under $\ell^2$ and $\ell^\infty$ it goes to zero; under $\ell^1$ it does not stir. “It approaches zero” is not a property of the sequence — it is a property of the sequence together with a ruler.
Observation 4.3 The equivalence theorem is not violated here. At each $n$ all three are equivalent — what breaks is moving $n$. The shape is that of Paper 211: the answer is not settled until one says which variable is being moved.
5. With Functions It Goes Both to Zero and to Infinity
On $[0,1]$ take the spike $f_n$ of height $n^{1/4}$ and width $1/n$.
Check 5.1:
| $n$ | Sup norm | $L^2$ norm | $L^1$ norm |
|---|---|---|---|
| $10$ | $1.7782794$ | $0.56234133$ | $0.17782794$ |
| $100$ | $3.1622777$ | $0.31622777$ | $0.031622777$ |
| $10^4$ | $10.0$ | $0.1$ | $0.001$ |
| $10^8$ | $\mathbf{100.0}$ | $\mathbf{0.01}$ | $\mathbf{10^{-6}}$ |
Observation 5.2 One sequence goes to infinity under one ruler and to zero under two others. This is no longer a question of a factor of disagreement — they disagree about whether it converges at all. And this is something only infinite dimension permits: in finite dimension it cannot happen.
6. Checked Numerically, Not Only by Formula
Check 6.1: integrate the spike on a grid of four million points and compare with the closed form.
| $n$ | Sup (grid) | $L^2$ (grid) | Closed form | Relative error |
|---|---|---|---|---|
| $10$ | $1.77827941$ | $0.5623413252$ | $0.5623413252$ | $\mathbf{0}$ |
| $100$ | $3.16227766$ | $0.3162277660$ | $0.3162277660$ | $\mathbf{0}$ |
Observation 6.2 The grid agrees with the closed form. The previous table is not a matter of algebra but numbers obtained by actually integrating.
7. The Unit Balls, Measured
Check 7.1: the volumes of the unit balls of $\ell^1,\ell^2,\ell^\infty$.
| $n$ | $\ell^1$ | $\ell^2$ | $\ell^\infty$ | $\ell^1/\ell^\infty$ |
|---|---|---|---|---|
| $2$ | $2.0$ | $3.1415927$ | $4.0$ | $0.5$ |
| $5$ | $0.26666667$ | $5.263789$ | $32.0$ | $0.0083333333$ |
| $10$ | $0.00028218695$ | $2.550164$ | $1024.0$ | $2.7557319\times10^{-7}$ |
| $50$ | $3.7019019\times10^{-50}$ | $1.7302192\times10^{-13}$ | $1.1258999\times10^{15}$ | $\mathbf{3.2879494\times10^{-65}}$ |
Observation 7.2 The $\ell^1$ ball is $1/n!$ of the $\ell^\infty$ ball. At $n=50$ that is one part in $\mathbf{3.04\times10^{64}}$ — two “equivalent” rulers whose unit balls differ in volume by sixty-four orders. And the $\ell^2$ ball itself has shrunk to $1.73\times10^{-13}$ by $n=50$: the curse of dimensionality that Paper 182 measured appears here as a curse on the choice of ruler.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the senses of “near” is which norm does the measuring. In finite dimension the choice does not change the answer — though by how much it does not worsens with the dimension. In infinite dimension the choice does change the answer — it changes whether there is convergence at all.
Observation 8.2 (the fence) The fence around “any ruler gives the same answer” is that the dimension be finite. On the other side of that same fence stands Riesz's lemma: the closed unit ball is compact only in finite dimension. The two are one fence — the place where the constant of the equivalence runs away is the place where the ball loses its compactness.
Observation 8.3 Paper 177 wrote that the order of the quantifiers is the whole content — the equivalence here is likewise a matter of whether the constant comes before the dimension or after. Fix $n$ and then look for a constant, and one is always found. Fix the constant first and look for one that serves every $n$, and there is none. One sentence turns from true to false by exchanging the order alone.
9. The Census and the Close
What this paper measured:
- The best constants are $\sqrt n$ and $n$ (Check 2.1).
- The extreme lies in one direction only (Check 2.2): $26.21$ against $31.62$ among $200000$.
- Equivalent, and the constant runs away (Check 3.2): six digits at $10^6$.
- One sequence does not stir under one ruler (Check 4.1): $\|x\|_1=1$ throughout.
- With functions, both to zero and to infinity (Check 5.1): $100$, $0.01$ and $10^{-6}$.
- Grid and closed form agree (Check 6.1).
- The unit ball is smaller by $3.04\times10^{64}$ (Check 7.1).
References
- Riesz, F. (1918). Über lineare Funktionalgleichungen. Acta Mathematica, 41.
- Banach, S. (1932). Théorie des opérations linéaires. Warsaw.
- Conway, J. B. (1990). A Course in Functional Analysis. Second edition. New York: Springer.
- Rudin, W. (1991). Functional Analysis. Second edition. New York: McGraw-Hill.
- Ball, K. (1997). An elementary introduction to modern convex geometry. In Flavors of Geometry. Cambridge: Cambridge University Press.
- Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer.
- Yamagishi, Y. (2026). The Order of the Quantifiers Is the Whole Content [Paper 177]. Zenodo. 10.5281/zenodo.22098103
- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
- Yamagishi, Y. (2026). “Converges” Is Four Separate Claims, and What Separates Them Is What You May Discard [Paper 197]. Zenodo. 10.5281/zenodo.22107590
- Yamagishi, Y. (2026). Curvature Does the Same Work as Dimension [Paper 208]. Zenodo. 10.5281/zenodo.22111071
- Yamagishi, Y. (2026). Adding More Terms Takes You Further from the Answer [Paper 211]. Zenodo. 10.5281/zenodo.22111772
- Yamagishi, Y. (2026). Coming Back Has Distinct Roots [Paper 213]. Zenodo. 10.5281/zenodo.22119380
- Yamagishi, Y. (2026). A Lens Can Brighten and Cannot Increase Brightness [Paper 214]. Zenodo. 10.5281/zenodo.22119392
Reference
Yamagishi, Y. (2026). One Sequence Goes Both to Zero and to Infinity [Paper 215]. YOI: https://www.yamagishiyuuki.com/yoi/215
DOI: 10.5281/zenodo.22119477
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 215. This paper is a preprint and has not been peer reviewed.