Pith Number

pith:AHIMJFC2

pith:2026:AHIMJFC2V2FPXOWDCWHVUQG3RZ

not attested not anchored not stored refs resolved

On weak convergence in K\"{o}the-Bochner function spaces

Jos\'e Rodr\'iguez

If X* lacks the Radon-Nikodým property, the closed unit ball of E*(X*) is not a James boundary for E(X).

arxiv:2605.13240 v1 · 2026-05-13 · math.FA

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{AHIMJFC2V2FPXOWDCWHVUQG3RZ}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

If X^* fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence (f_n) in E(X) such that ⟨ϕ,f_n⟩→0 for every ϕ∈E^*(X^*); in particular, the closed unit ball of E^*(X^*) is not a James boundary for E(X).

C2weakest assumption

E is an order continuous Köthe function space over a non purely atomic probability measure μ (stated in the opening sentence of the abstract).

C3one line summary

If X* fails the Radon-Nikodým property then the closed unit ball of E*(X*) is not a James boundary for the Köthe-Bochner space E(X) when E is order continuous over a non-purely-atomic measure.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] P. A. H. Brooker,Non-Asplund Banach spaces and operators, J. Funct. Anal.273(2017), no. 12, 3831–3858 2017

[2] B. Cascales and A. J. Pallar´ es,La propiedad de Radon-Nikodym en espacios de Banach duales, Collect. Math.45(1994), no. 3, 263–270 1994

[3] J. Diestel and J. J. Uhl, Jr.,Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977 1977

[4] Dwivedi,Weak ∗-weak points of continuity on the state spaces, Rev 2026

[5] G. A. Edgar,Asplund operators and a.e. convergence, J. Multivariate Anal.10(1980), no. 3, 460–466 1980

Receipt and verification

First computed	2026-05-18T02:44:49.512028Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

01d0c4945aae8afbbac3158f5a40db8e752a91d7c568c4378048fd8085346ec3

Aliases

arxiv: 2605.13240 · arxiv_version: 2605.13240v1 · doi: 10.48550/arxiv.2605.13240 · pith_short_12: AHIMJFC2V2FP · pith_short_16: AHIMJFC2V2FPXOWD · pith_short_8: AHIMJFC2

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/AHIMJFC2V2FPXOWDCWHVUQG3RZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 01d0c4945aae8afbbac3158f5a40db8e752a91d7c568c4378048fd8085346ec3

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8280a233cf1b65b8083ac40c612b4149f47ff7d9e6237ffa38d8b29aa1f34b08",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-13T09:26:14Z",
    "title_canon_sha256": "1c0b69d7ff1d11563cfa407bfdbdb78268c03f0b7f1b1e6f0193aa1ef477bc3d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13240",
    "kind": "arxiv",
    "version": 1
  }
}