On the spaceability of the set of functions in the Lebesgue space \(L_p\) which are in no other \(L_q\)

In this note we prove that, for \(p>0\), \(L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]\) is \((\alpha,\mathfrak{c})\)-spaceable if, and only if, \(\alpha<\aleph_{0}\). Such a problem first appears in [V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020)...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Araújo, Gustavo, Anderson, Barbosa, Raposo, Anselmo, Ribeiro, Geivison
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.04.2023
Online AccessGet full text

Cover

More Information
Summary:In this note we prove that, for \(p>0\), \(L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]\) is \((\alpha,\mathfrak{c})\)-spaceable if, and only if, \(\alpha<\aleph_{0}\). Such a problem first appears in [V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the \((1,\mathfrak{c})\)-spaceability of \(L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]\) for \(p>0\). The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.
ISSN:2331-8422