REVIEW 3 major objections 5 minor 23 references
Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For nuclear spaces with DN and Ω, diametral-dimension equality with Λ∞(ε) is equivalent to approximate-diametral-dimension equality; in the finite-type case, the equivalence holds when a prominent bounded set exists.
desk verdict A worthwhile question, but Lemma 3.2 is wrong, so the infinite-type main theorem does not hold as written; the finite-type section has salvageable pieces but needs serious rework. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing machinery is the exponential comparison between Kolmogorov diameters and the exponent sequence ε. Writing ε_n(p,q) = -log d_n(U_q,U_p), the relation Δ(E)=Δ(Λ∞(ε)) is shown to be equivalent to the condition inf_p sup_{q≥p} liminf ε_n(p,q)/ε_n = +∞, while δ(E)=δ(Λ1(ε)) is characterized by inf_p sup_{q≥p} limsup ε_n(p,q)/ε_n = 0. The finite-type results add the notion of a prominent bounded set—an absolutely convex bounded set B such that the diameters d_n(B,U_p) alone determine Δ(E)—and show it is equivalent to the Dragilev-type condition D2, which in turn implies the barrelledness condition (wQ) needed for the canonical-topology argument.
What would settle it
For E=Λ∞(ε) itself, compute δ(E) from the definition and from the Lemma 3.2 sup-condition formula; if the sup-condition spaces omit sequences that vanish in the corresponding lim ratio, the representation is not the true approximate diametral dimension and the necessity proof of Theorem 3.1 collapses.
Extended reading notes
Core claim
The central discovery is that the transfer question has an affirmative answer in the infinite-type case and a conditional affirmative answer in the finite-type case. Theorem 3.1 states that for a nuclear Fréchet space E with DN and Ω and associated exponent sequence ε, Δ(E)=Δ(Λ∞(ε)) if and only if δ(E)=δ(Λ∞(ε)). In the finite-type case, Proposition 4.1 shows δ(E)=δ(Λ1(ε)) always implies Δ(E)=Δ(Λ1(ε)); Theorem 4.8 shows the converse holds precisely when E has a prominent bounded set. The paper also records that the same conclusion follows if Δ(E) is barrelled in its canonical topology (Theorem 4.2), and that a prominent bounded set is equivalent to the diameter condition D2 (Proposition 4.5).
Load-bearing premise
The infinite-type result depends on representing the approximate diametral dimension as a union-over-p, intersection-over-q family of spaces defined by a supremum condition and assuming this representation has the topological property needed for a closed-graph argument; neither the representation nor that property is proved.
Editorial extensions
If this is right
- For infinite-type spaces, the two invariants carry identical information: to establish Δ(E)=Δ(Λ∞(ε)) it is enough and necessary to establish δ(E)=δ(Λ∞(ε)).
- In finite type, approximate-dimension equality is the stronger condition: it automatically gives diametral-dimension equality and forces the existence of a prominent bounded set.
- A nuclear space with DN and Ω whose diametral dimension is barrelled in its canonical topology gets the full finite-type equivalence without checking prominence separately.
- The prominent-bounded-set condition is equivalent to the diameter ratio condition D2, so the finite-type theorem can be tested directly from ratios d_n(U_q,U_p)/d_n(U_k,U_q).
- If Δ(E) is closed under coordinatewise multiplication, then Δ(E) equals the auxiliary class Φ(E); combined with prominence this gives δ(E)=δ(Λ1(ε)).
Reading between the lines
- The Lemma 3.2 representation of δ(E) as an inductive limit of Banach spaces with a supremum condition is the least protected step in the infinite-type proof; a direct proof of Theorem 3.1 that avoids this representation would put the equivalence on firmer ground.
- The finite-type converse may fail outside the prominent-bounded-set hypothesis; looking for a DN-Ω nuclear space with Δ(E)=Δ(Λ1(ε)) but δ(E)≠δ(Λ1(ε)) would either confirm the necessity of prominence or produce a counterexample that sharpens Theorem 4.8.
- For concrete spaces, the theorem turns a structural question into a geometric one: to know whether tameness holds, check whether the diametral dimension of E is generated by a single bounded set.
- The algebra criterion offers a practical route: proving Δ(E) is closed under squares is enough, in the presence of a prominent bounded set, to conclude δ(E)=δ(Λ1(ε)).
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies whether, for a nuclear Fréchet space E with properties DN and Ω, equality of the diametral dimension Δ(E) with that of a power series space forces equality of the approximate diametral dimension δ(E), and vice versa. For infinite-type power series spaces, Theorem 3.1 asserts that Δ(E)=Δ(Λ∞(ε)) holds if and only if δ(E)=δ(Λ∞(ε)). For finite-type spaces, Proposition 4.1 proves that δ(E)=δ(Λ1(ε)) implies Δ(E)=Δ(Λ1(ε)), and Theorem 4.8 claims an equivalence under the additional assumption that E has a prominent bounded subset. Auxiliary results include characterizations involving condition (wQ), conditions A and B, and a relation between prominent bounded sets and Dragilev's condition D2 (Proposition 4.5).
Significance. The question addressed is natural and relevant to the structure theory of nuclear Fréchet spaces, where diametral and approximate diametral dimensions are standard invariants. If the results were correct, they would provide clean dichotomies connecting the associated exponent sequence, prominent bounded sets, and the invariants of power series spaces. The paper is clearly written and draws on appropriate references, including work of Aytuna, Krone, and Terzioğlu and of Demeulenaere, Frerick, and Wengenroth. However, the central infinite-type theorem relies on a set-theoretic identification in Lemma 3.2 that is false as stated, and the finite-type results rely on an unproved and in general invalid equivalence in Proposition 4.5. These are load-bearing gaps, so the advertised results are not established by the manuscript.
major comments (3)
- [Section 3, Lemma 3.2] Lemma 3.2 identifies δ(E) with the union over p of intersections over q≥p of the Banach spaces δ_{pq} = { (t_n): sup_n |t_n|/d_n(U_q,U_p) < ∞ }. This contradicts the definition of δ(E) given in Section 2, which is { (t_n): ∃p ∀q>p lim_n |t_n|/d_n(U_q,U_p)=0 }. Boundedness of the ratio does not imply that the ratio tends to 0, and the q=p factor in the intersection is degenerate because d_n(U_p,U_p)=1 for an infinite-dimensional Hilbert space, so that factor only imposes boundedness of (t_n). Both directions of Theorem 3.1 pass through Lemma 3.2, so the infinite-type equivalence is not established as written.
- [Section 4, Proposition 4.5] The proof of the equivalence 2⇔3 only establishes 2⇒3; the converse direction is asserted without argument. The condition sup_{l≥q} limsup_n ε_n(q,l)/ε_n(p,q) ≤ 1 does not by itself imply D2: for example, if ε_n(q,l)=ε_n(p,q)+log n for all l≥q and ε_n(p,q)→∞, then the limsup ratio is 1, but d_n(U_q,U_p)/d_n(U_l,U_q) = e^{ε_n(q,l)-ε_n(p,q)} = n → ∞, so the limit required in D2 fails. Since Corollary 4.7 and Theorem 4.8 rely on the prominent-bounded-set/D2 equivalence, the finite-type main results are not supported.
- [Section 4, Theorem 4.2 and Proposition 4.3] In the sufficiency part of Theorem 4.2 and in the contradiction argument of Proposition 4.3, continuity of inclusions of the form ∩_p ∪_q Δ(U_q,U_p) ↪→ Λ1(ε) is converted into pointwise inequalities such as e^{-tε_n} ≤ C d_n(U_q,U_p) for all n. This conversion assumes that a continuous seminorm on a projective limit of inductive limits of Banach spaces is dominated by the norm of a single Banach step; that is not true for arbitrary non-regular LB-spaces, and no regularity argument is supplied. Consequently the deductions of δ(E)=δ(Λ1(ε)) in these results are not justified.
minor comments (5)
- [Section 2] The sup representation of Δ(E) attributed to [8] should be stated with its precise hypotheses; in general, the condition sup_n |t_n|d_n(U_q,U_p)<∞ is weaker than lim_n |t_n|d_n(U_q,U_p)=0, so the equivalence is not automatic and the reader needs the exact theorem being cited.
- [Section 2, Theorem 2.1] The statement of condition (wQ) is hard to read because n is used both as an index of the seminorm and as the sequence index in the diameters; please disambiguate the notation.
- [Sections 3 and 4] The indexing in Lemma 3.2 uses q≥p, whereas the definition of δ(E) in Section 2 uses q>p; even apart from the set-equality problem, the degenerate q=p term should be addressed explicitly.
- [Theorem 4.8 and Proposition 4.3] The notation Δ(E)=Λ1(ε) should be Δ(E)=Δ(Λ1(ε)); as printed, it suggests equality of a sequence space with a Fréchet space rather than with its diametral dimension.
- [Throughout] There are several typographical errors, including 'refeer' (p. 3), 'Thoughout' (p. 4), 'nuclaer' (Corollary 4.10), and inconsistent spelling of 'Terzioğlu'.
Circularity Check
No circularity found: the main equivalences are proved from external published theorems, with no fitted inputs or self-referential definitions; the Lemma 3.2 issue is a correctness gap, not a circular reduction.
full rationale
The derivation chain in this paper is not circular. The main theorems (3.1, 4.2, 4.8) are proved using external results: the inclusion and associated-exponent-sequence facts of Aytuna-Krone-Terzioğlu [3], Aytuna's characterization (3.1) from [5], standard barrelledness/closed-graph arguments, and the prominent-bounded-set results of Djakov-Terzioğlu [9]. None of these is a restatement of the paper's target equalities, and none is fitted to the data being 'predicted.' The paper does not define Δ(E) or δ(E) in terms of the power-series equalities it proves, nor does it equip δ(E) with a topology that secretly assumes the conclusion; the barrelledness arguments use standard theorems about inductive limits. Proposition 2.3 is cited from [3], whose authors do not include the present paper's author; the reliance on Aytuna [5] is heavy, but that is an external published characterization, not a self-citation chain that forces the conclusions. The most serious issue is Lemma 3.2, where the text writes δ(E)=⋃_p⋂_{q≥p}δ_{pq} with δ_{pq} defined by a sup-condition, while Section 2 defines δ(E) by a lim-condition; this is a mathematical gap in the proof of Theorem 3.1, but it is not a circular reduction—the sup-space is not defined in terms of the theorem's conclusion, and no parameter is fitted or renamed as a prediction. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (5)
- domain assumption The associated exponent sequence epsilon of E is such that Lambda_1(epsilon) is nuclear.
- ad hoc to paper delta(E) equals the union over p of the intersection over q of delta_{pq} with delta_{pq} = {sup_n |t_n|/d_n(U_q,U_p) < infinity}, and this LB-space is barrelled.
- ad hoc to paper D2, prominence of a bounded set, and the ratio condition sup_l limsup epsilon_n(q,l)/epsilon_n(p,q) <= 1 are equivalent.
- ad hoc to paper Conditions A and B imply the conclusion of Proposition 4.3 and 4.4 via the constructed projective limit embeddings.
- standard math Known characterizations of Delta and delta for power series spaces and the diameter estimates from Proposition 2.3 hold as stated.
Cite this review
Pith. "Pith review of Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces." pith.science (2026). https://pith.science/paper/NQCXDO36
@misc{pith2026190801838,
author = {Pith},
title = {Pith review of: Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces},
year = {2026},
howpublished = {\url{https://pith.science/paper/NQCXDO36}},
note = {Machine review of arXiv:1908.01838}
}
abstract
The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta (E)$, of a nuclear Fr\'echet space $E$ which satisfies $\underline{DN}$ and $\Omega$, is related to power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. In this article, we examine a question of whether $\delta (E)$ must coincide with that of a power series space if $\Delta(E)$ does the same, and vice versa. In this regard, we first show that this question has an affirmative answer in an infinite type case by proving the fact that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if $\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.
Reference graph
Works this paper leans on
-
[1]
S. Agethen, K. D. Bierstedt, and J. Bonet, Projective limits of weighted (LB)- spaces of continuous functions, Arch. Math. (Basel) 92 (2009), no. 5, 384398
work page 2009
- [2]
- [3]
- [4]
-
[5]
Aytuna, Tameness in Fr´ echet spaces of analytic functions, Studia Math., 232 (2016), 243-266
A. Aytuna, Tameness in Fr´ echet spaces of analytic functions, Studia Math., 232 (2016), 243-266
work page 2016
-
[6]
Bessaga, Some remarks on Dragilev’s theorem , Studia Math
Cz. Bessaga, Some remarks on Dragilev’s theorem , Studia Math. 31 (1968), 307-318
work page 1968
-
[7]
Cz. Bessaga, A. Pelczynski and S. Rolewicz, On diametral approximative dimension and linear ho- mogeneity of F-spaces. Bull. Acad. Pol. Sci, 9, 307-318, (1961)
work page 1961
-
[8]
L. Demeulenaere, L. Frerick, J. Wengenroth, Diametral dimensions of Fr´ echet spaces, Studia Math. 234 (3) (2016) 271-280. 13
work page 2016
Show all 23 references
-
[9]
Djakov, T
P. Djakov, T. Terzio˘ glu,Prominent Sets in Fr´ echet Spaces, preprint
-
[10]
Dubinsky, The structure of nuclear Fr´ echet spaces, LNM, Springer, 720, (1979)
E. Dubinsky, The structure of nuclear Fr´ echet spaces, LNM, Springer, 720, (1979)
1979
-
[11]
Husain, The Open Mapping and Closed Graph Theorems in Topological Vector Spaces , Springer-Verlag, (1965)
T. Husain, The Open Mapping and Closed Graph Theorems in Topological Vector Spaces , Springer-Verlag, (1965)
1965
-
[12]
Kolmogorov, On the linear dimension of topological vector spaces, Dokl
A.N. Kolmogorov, On the linear dimension of topological vector spaces, Dokl. Akad. Nauk SSSR 120, No. 2 (1958), 239-241
1958
-
[13]
K¨ othe,Topological Vector Spaces I
G. K¨ othe,Topological Vector Spaces I. Springer-Verlag, 1969
1969
-
[14]
Meise and D
R. Meise and D. Vogt, Introduction to functional analysis , Clarendon Press, Oxford, 1997
1997
-
[15]
Mityagin, Geometry of nuclear spaces
B.S. Mityagin, Geometry of nuclear spaces. II - Linear topological invariants S´ eminaire d’analyse fonctionnelle (Polytechnique) (1978-1979), exp. no 2, p. 1-10
1978
-
[16]
Mityagin, Approximative dimension and bases in nuclear spaces
B.S. Mityagin, Approximative dimension and bases in nuclear spaces. Russian Math. Surveys 16(1961), 59127
1961
-
[17]
Pelczynski, On the approximation of S-spaces by finite-dimensional spaces , Bull
A. Pelczynski, On the approximation of S-spaces by finite-dimensional spaces , Bull. Acad. Pol. Sci
-
[18]
9 (1957), 879-881
No. 9 (1957), 879-881
1957
-
[19]
Pietsch, Nuclear Locally Convex Spaces, Ergebnisse der Math
A. Pietsch, Nuclear Locally Convex Spaces, Ergebnisse der Math. 66, Springer-Verlag, 1972
1972
-
[20]
Rolewicz, Metric linear spaces 2nd ed., Mathematics and its Applications (East European Series), vol
S. Rolewicz, Metric linear spaces 2nd ed., Mathematics and its Applications (East European Series), vol. 20, D. Reidel Publishing Co., Dordrecht; PWNPolish Scientific Publishers, Warsaw, 1985
1985
-
[21]
Terzio˘ glu,Die diametrale Dimension von lokalkonvexen R¨ aumen, Collect
T. Terzio˘ glu,Die diametrale Dimension von lokalkonvexen R¨ aumen, Collect. Math. 20(1969), 49-99
1969
-
[22]
Terzio˘ glu, Some invariants of Fr´ echet spaces and imbeddings of smooth sequence spaces , T.Terzio˘ glu (ed.), Advances in the theory of Fr´ echet spaces, 305-324 (1989)
T. Terzio˘ glu, Some invariants of Fr´ echet spaces and imbeddings of smooth sequence spaces , T.Terzio˘ glu (ed.), Advances in the theory of Fr´ echet spaces, 305-324 (1989)
1989
-
[23]
Terzio˘ glu,Quasinormability and diametral dimension, Turkish J
T. Terzio˘ glu,Quasinormability and diametral dimension, Turkish J. Math.37 (2013), no. 5, 847-851. 14
2013
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