Abstract
Let m be an ¿ 2 -valued (countably additive) vector measure and consider the space L 2 (m) of square integrable functions with respect to m. The integral with respect to m allows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of strongly m-orthonormal sequences. Combining the use of the Kadec-Pelczyski dichotomy in the domain space and the Bessaga-Pelczyski principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces L 1 (m) and L 2 (m). Under certain requirements, our main result establishes that a normalized sequence in L 2 (m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in L 2 (m *), where m * is another ¿ 2 -valued vector measure that satisfies L 2 (m) = L 2 (m *). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an ¿ 2 -valued positive vector measure contains a lattice copy of ¿ 2. © 2012 E. Jiménez Fernández and E. A. Sánchez Pérez.
