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A Counterexample to Two Compactness Conjectures of Brudnyi

  • Jinghao Huang
  • Ran Xu
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Abstract

We construct a quasi-Banach function lattice X on ℝ whose quasi-norm is order continuous and invariant under every translation. Nevertheless, X contains a bounded sequence with common compact support which is equicontinuous in X and has no norm-convergent subsequence. This provides a counterexample to Conjectures 2.5 and 2.11 proposed by Brudnyi in \cite{Brudnyi2015}.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026counterexample,
  title = {{A Counterexample to Two Compactness Conjectures of Brudnyi}},
  author = {Jinghao Huang and Ran Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11066},
  eprint = {2610.11066},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11066},
}
RIS
TY  - UNPB
TI  - A Counterexample to Two Compactness Conjectures of Brudnyi
AU  - Jinghao Huang
AU  - Ran Xu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11066
UR  - https://arxiv.org/abs/2610.11066
AB  - We construct a quasi-Banach function lattice X on ℝ whose quasi-norm is order continuous and invariant under every translation. Nevertheless, X contains a bounded sequence with common compact support which is equicontinuous in X and has no norm-convergent subsequence. This provides a counterexample to Conjectures 2.5 and 2.11 proposed by Brudnyi in \cite{Brudnyi2015}.
ER  -