Subspaces of maximal dimension contained in $L_{p}(Ω) - \textstyle\bigcup\limits_{q<p}L_{q}(Ω)$}

Details Subspaces of maximal dimension contained in $L_{p}(Ω) - \textstyle\bigcup\limits_{q<p}L_{q}(Ω)$} Subspaces of maximal dimension contained in $L_{p}(Ω) - \textstyle\bigcup\limits_{q<p}L_{q}(Ω)$}

2012-04-10

arxiv.org/abs/1204.2170v1

Mathematics

Functional Analysis

Scientific paper

Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we determine when the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, when it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$ such that $\dim(F) = \dim(L_{p}(\Omega))$. 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 $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$. We shall also give examples, propose open questions and provide new directions in the study of maximal subspaces of classical measure spaces.

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