Abstract
“It approaches” can mean that the distance shrinks or that every test stops seeing a difference. This paper separates the two in numbers and measures a sequence for which only the second happens and the first never does. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, an orthonormal sequence $e_n$ tends to zero against every fixed test. Against $y_k=1/k$ it falls as $1/n$ and against $y_k=2^{-k}$ as $2^{-n}$ — and $\|e_n\|$ is exactly $1$ throughout (Section 2).
Second, no finite set of tests can ever see it. In $\mathbb{R}^{1000}$ with fifty tests one can still find a vector of length $1$ on which every reading is $10^{-15}$ (Section 3).
Third, the same holds for functions. The $L^2$ size of $f_n(x)=\sin(2\pi nx)$ is $0.7071067812$ at every $n$, while the test integrals fall off as one over $n$ (Section 4).
Fourth, this is the core. Although $f_n$ tends weakly to zero, the test integrals of $f_n^2$ go to $0.859140831213$, which is $\tfrac12\int e^x=0.859140914230$ — the square of the weak limit is $0$ and the weak limit of the square is half the average (Section 5).
Fifth, in finite dimension a unit vector cannot hide. A vector of length $1$ in $\mathbb{R}^N$ must read at least $1/\sqrt N$ on some coordinate (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — weak convergence, the Riemann–Lebesgue lemma, the Radon–Riesz property and the Banach–Alaoglu theorem are all standard. The weak topology is not constructed — what is treated is sequences, not the topology itself. The theory of compensated compactness is not developed — Section 5 touches its simplest instance only. Neither dual spaces nor reflexivity is treated. Nothing is claimed about homogenisation or turbulence closure. On a name: the corpus already carries a theorem of Riesz, in Paper 197, and that one says that convergence in measure yields an almost-everywhere convergent subsequence — the Riesz here is that weak convergence together with convergence of the norms is strong convergence, a different theorem by the same man. Relation to Paper 215: what 215 varied was which ruler does the measuring — what is varied here is what counts as a test, with the ruler held fixed. The fence is the same one, that the dimension be finite, and what fails at it is different. What is added is setting the test values and the size of an orthonormal sequence side by side, actually constructing a vector invisible to finitely many tests, measuring that the size of an oscillating sequence does not move while its test integrals do, confirming with three test functions that the weak limit of the square is half the average, measuring that the difference vanishes once the norms agree, and confirming by search that the floor in finite dimension is $1/\sqrt N$.
1. Notation and Position
For a sequence $x_n$ in an inner-product space, approaching strongly means $\|x_n-x\|\to0$, and approaching weakly means $\langle x_n,y\rangle\to\langle x,y\rangle$ for every fixed $y$. Strong implies weak, and the converse fails.
Position: Paper 197 separated the four roots of “converges” by what one is allowed to discard — what is separated here is what one goes to look at. In Paper 215 “near” depended on the choice of ruler — here the ruler is one and the ways of looking are two.
2. An Orthonormal Sequence: Tests to Zero, Size Fixed at One
Check 2.1: let $e_n$ be the $n$-th coordinate vector of $\ell^2$ and test it against two fixed vectors.
| $n$ | Test against $y_k=1/k$ | Test against $y_k=2^{-k}$ | $\|e_n\|$ |
|---|---|---|---|
| $1$ | $1.0$ | $0.5$ | $1.0$ |
| $5$ | $0.2$ | $0.03125$ | $1.0$ |
| $20$ | $0.05$ | $9.5367432\times10^{-7}$ | $1.0$ |
| $100$ | $0.01$ | $7.8886091\times10^{-31}$ | $1.0$ |
| $1000$ | $\mathbf{0.001}$ | $\mathbf{9.3326362\times10^{-302}}$ | $\mathbf{1.0}$ |
Observation 2.2 Every fixed test goes to zero and the size never moves once. $\|e_n-0\|=1$ for ever — “it is approaching zero” and “it stays at distance one from zero” are both true of the same sequence.
3. No Finite Set of Tests Can See It
Check 3.1: in $\mathbb{R}^n$ take $k$ random tests and look for a vector of length $1$ on which every test reads zero.
| $n$ | $k$ | Largest reading | Length of the vector |
|---|---|---|---|
| $10$ | $5$ | $6.661\times10^{-16}$ | $1.000000000000$ |
| $100$ | $5$ | $3.886\times10^{-16}$ | $1.000000000000$ |
| $1000$ | $5$ | $6.661\times10^{-16}$ | $1.000000000000$ |
| $1000$ | $50$ | $\mathbf{1.887\times10^{-15}}$ | $\mathbf{1.000000000000}$ |
Observation 3.2 So long as the dimension exceeds the number of tests some direction is invisible to them. Weak convergence is that loophole taken to the limit — the tests may be infinite in number, but they are used one fixed at a time, so there is always a direction not yet looked at.
4. The Same with Functions: Faster and Faster Oscillation
Check 4.1: take $f_n(x)=\sin(2\pi nx)$ on $[0,1]$.
| $n$ | $\|f_n\|_{L^2}$ | $\int f_n\,x$ | $\int f_n\,e^x$ | $\int f_n/(1+x)$ |
|---|---|---|---|---|
| $1$ | $0.7071067812$ | $-1.592\times10^{-1}$ | $-2.667\times10^{-1}$ | $7.401\times10^{-2}$ |
| $16$ | $0.7071067812$ | $-9.947\times10^{-3}$ | $-1.709\times10^{-2}$ | $4.972\times10^{-3}$ |
| $64$ | $0.7071067812$ | $-2.487\times10^{-3}$ | $-4.273\times10^{-3}$ | $1.243\times10^{-3}$ |
| $256$ | $\mathbf{0.7071067812}$ | $\mathbf{-6.217\times10^{-4}}$ | $\mathbf{-1.068\times10^{-3}}$ | $\mathbf{3.108\times10^{-4}}$ |
($1/\sqrt2=0.7071067812$; the integrals use a grid of $2^{22}$ points.)
Observation 4.2 The test integrals fall as one over $n$ (Riemann–Lebesgue) and the size does not move to ten decimal places. The wave only gets faster and does not get smaller — what is going to zero is not the wave but its engagement with a smooth partner.
5. The Core: Squaring Moves the Limit
Check 5.1: test $f_n^2$ for the same $f_n$.
| $n$ | $\int f_n^2\,x$ | $\int f_n^2\,e^x$ | $\int f_n^2/(1+x)$ |
|---|---|---|---|
| $1$ | $0.250000000000$ | $0.853734576864$ | $0.344299547225$ |
| $16$ | $0.250000000000$ | $0.859119662515$ | $0.346564315769$ |
| $256$ | $\mathbf{0.250000000000}$ | $\mathbf{0.859140831213}$ | $\mathbf{0.346573554045}$ |
| $\tfrac12\int g$ | $0.250000000000$ | $0.859140914230$ | $0.346573590280$ |
| Square of the weak limit | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |
Observation 5.2 The weak limit of the square is not the square of the weak limit. $f_n\to0$ weakly and $f_n^2\to\tfrac12$ weakly — not $0$ but half the average. All three test functions agree with $\tfrac12\int g$ to seven decimal places.
Observation 5.3 Here is the price of approaching weakly. Strong convergence carries squares along with it; weak convergence passes only what is linear — and the square is the energy, the variance, the intensity. So only quantities of first order may be called gone, and those of second order remain.
Observation 5.4 One can even say where what seemed to vanish has gone — the oscillation merely got faster, and the energy is constant. This paper does not go on from here to homogenisation or turbulence closure: only the simplest instance has been touched.
6. Once the Norms Agree, Weak Becomes Strong
Check 6.1: $x_n=(1-1/n)e_1+(1/n)e_2$ tends weakly to $e_1$.
| $n$ | $\|x_n\|$ | $\|e_1\|$ | $\|x_n-e_1\|$ |
|---|---|---|---|
| $2$ | $0.707106781187$ | $1.0$ | $0.707106781187$ |
| $10$ | $0.905538513814$ | $1.0$ | $0.141421356237$ |
| $100$ | $0.990050503762$ | $1.0$ | $0.014142135624$ |
| $10000$ | $\mathbf{0.999900005001}$ | $1.0$ | $\mathbf{0.000141421356}$ |
Observation 6.2 Once the sizes settle onto the size of the limit, the difference goes to zero (Riesz). For the orthonormal sequence of Section 2 the size stays at $1$ and does not settle onto the weak limit's size of $0$ — so the difference does not go to zero either. Adding one single thing to weak convergence makes it strong, and that thing is that the sizes agree.
7. In Finite Dimension a Unit Vector Cannot Hide
Check 7.1: put a vector of length $1$ in $\mathbb{R}^N$ through the $N$ coordinate tests. How small can the largest reading be made?
| $N$ | Least possible, $1/\sqrt N$ | Least among $4000$ random |
|---|---|---|
| $1$ | $1.0$ | $1.0$ |
| $4$ | $0.5$ | $0.51812550$ |
| $100$ | $0.1$ | $0.19566903$ |
| $10000$ | $\mathbf{0.01}$ | $0.03278663$ |
Observation 7.2 The floor is $1/\sqrt N$, and it is positive at every finite $N$. A vector of length $1$ must show itself on some test — the floor reaches zero only where $N$ does not stop, and that is where weak and strong part. Random directions do not reach the floor ($0.0328$ against $0.01$ at $N=10^4$): it is reached only by the all-equal direction.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the two senses of approaching is what one goes to look at.
| Phrase | Demand | For $e_n$ |
|---|---|---|
| Approaches strongly | the distance goes to zero | does not happen (stays $1$) |
| Approaches weakly | every fixed test loses the difference | happens |
| Weakly, and the sizes agree | both of the above | does not happen |
Observation 8.2 (the fence) The fence around “approaching means getting nearer” is that the dimension be finite. It is the same fence as in Paper 215 with a different thing failing at it: there it was which ruler is chosen that began to matter, and here it is that finitely many tests no longer suffice. Outside one fence, several distinct failures stand side by side.
Observation 8.3 And one more. Under weak convergence only linear quantities are carried to the limit — and the moment one squares, $0$ turns into $\tfrac12$. That taking the limit and applying a function cannot be exchanged is exactly the price of having left strong convergence. In the language of Paper 177 this too was a question of order.
9. The Census and the Close
What this paper measured:
- Tests to zero, size fixed at one (Check 2.1).
- Invisible to finitely many tests (Check 3.1): length $1$ even with fifty tests in a thousand dimensions.
- Oscillation does not shrink (Check 4.1): fixed at $0.7071067812$.
- The weak limit of the square is half the average (Check 5.1): $0.859140831213$ against $0.859140914230$.
- Once the sizes agree the difference goes (Check 6.1): $1.41\times10^{-4}$.
- The floor in finite dimension is $1/\sqrt N$ (Check 7.1).
References
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- 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). Adding More Terms Takes You Further from the Answer [Paper 211]. Zenodo. 10.5281/zenodo.22111772
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Reference
Yamagishi, Y. (2026). It Approaches and Has Not Come One Step Closer [Paper 216]. YOI: https://www.yamagishiyuuki.com/yoi/216
DOI: 10.5281/zenodo.22119579
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 216. This paper is a preprint and has not been peer reviewed.