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

TL;DR

The paper proves that for every p>0, the set of functions in L_p[0,1] not belonging to any L_q[0,1] for q>p contains an infinite-dimensional closed linear subspace, answering a question from 2010.

Contribution

It establishes the spaceability of the set of functions in L_p[0,1] outside all larger L_q spaces for p>0, extending previous results and solving an open problem.

Findings

Existence of infinite-dimensional closed subspace within L_p[0,1] outside all L_q for q>p

Positive answer to a 2010 question on spaceability of these sets

Extension of results to sequence spaces

Abstract

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.