Distinguished subspaces of $L_p$ of maximal dimension
Volume 215 / 2013
Abstract
Let $(\varOmega ,\varSigma ,\mu )$ be a measure space and $1< p < \infty $. We show that, under quite general conditions, the set $L_{p}(\varOmega ) - \bigcup _{1 \leq q < p}L_{q}(\varOmega )$ is maximal spaceable, that is, it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\varOmega )$ such that $\operatorname{dim}(F) = \operatorname{dim}(L_{p}(\varOmega ))$. This result is so general that we had to develop a hybridization technique for measure spaces in order to construct a space such that the set $L_p(\varOmega ) - L_q(\varOmega )$, $1 \leq q < p$, fails to be maximal spaceable. In proving these results we have computed the dimension of $L_p(\varOmega )$ for arbitrary measure spaces $(\varOmega ,\varSigma ,\mu )$. 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 $L_{p}(\varOmega ) - L_{q}(\varOmega )$ with $q < p$ and $L_{p}(\varOmega ) - \bigcup_{1 \leq q < p}L_{q}(\varOmega )$.
