$L_{p}[0,1] \setminus \bigcup\limits_{q>p} L_{q}[0,1]$ is spaceable for every $p>0$

D. Pellegrino; G. Botelho; J. B. Seoane-Sep\'ulveda; V. V. F\'avaro

arxiv: 1106.0309 · v1 · pith:HW65NBY4new · submitted 2011-06-01 · 🧮 math.FA

L_(p)[0,1] setminus bigcuplimits_(q>p) L_(q)[0,1] is spaceable for every p>0

G. Botelho , V. V. F\'avaro , D. Pellegrino , J. B. Seoane-Sep\'ulveda This is my paper

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keywords everybigcuplimitsaboveanswersaronbanachbdfp

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In this short note we prove the result stated in the title; that is, for every $p>0$ there exists an infinite dimensional closed linear subspace of $L_{p}[0,1]$ every nonzero element of which does not belong to $\bigcup\limits_{q>p} L_{q}[0,1]$. This answers in the positive a question raised in 2010 by R. M. Aron on the spaceability of the above sets (for both, the Banach and quasi-Banach cases). We also complete some recent results from \cite{BDFP} for subsets of sequence spaces.

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L_(p)[0,1] setminus bigcuplimits_(q>p) L_(q)[0,1] is spaceable for every p>0

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