Distinguished subspaces of L-p of maximal dimension

Abstract

Let (Omega, Sigma, mu) be a measure space and 1 < p < infinity. We show that, under quite general conditions, the set L-p(Omega) - boolean OR(1 <= 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)) 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(Omega) - L-q (Omega),1 <= q < p, fails to be maximal spaceable. In proving these results we have computed the dimension of L-p(Omega) for arbitrary measure spaces (Omega, Sigma, 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(Omega) - L-q(Omega) with q < p and L-p(Omega) - boolean OR(1 <= q<p) L-q(Omega).