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)...
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Published in | arXiv.org |
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Main Authors | , , , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
19.04.2023
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Online Access | Get full text |
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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. |
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ISSN: | 2331-8422 |