Subspaces of maximal dimension contained in $L_p(\Omega) - \bigcup\limits_{q<p} L_q (\Omega)$

TL;DR

This paper demonstrates that, under broad conditions, the set of functions in $L_p(\Omega)$ not in any smaller $L_q(\Omega)$ space is maximally spaceable, extending classical results on the linear structure of these function spaces.

Contribution

It establishes maximal spaceability of certain difference sets in $L_p$ spaces under general conditions, generalizing decades of previous work.

Findings

The set $L_p(\Omega) - igcup_{q<p} L_q(\Omega)$ is maximal spaceable.

Failure of conditions can lead to non-maximal spaceability.

Provides a broad generalization of classical results on the structure of $L_p$ spaces.

Abstract

Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we show that, under quite general conditions, the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$ such that $\dim(F) = \dim(L_{p}(\Omega))$. We also show that if those conditions are not fulfilled, then even the larger set $L_p(\Omega) - L_q(\Omega)$, $1 \leq q < p$, may fail to be maximal spaceable. The aim of the results presented here is, among others, to generalize all the previous work (since the 1960's) related to the linear structure of the sets $L_{p}(\Omega) - L_{q}(\Omega)$ with $q < p$ and $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$.