Letmbe anℓ2-valued (countably additive) vector measure and consider the spaceL2(m) of square integrable functions with respect tom. The integral with respect tomallows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of stronglym-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spacesL1(m) andL2(m). Under certain requirements, our main result establishes that a normalized sequence inL2(m) with a weakly null sequence of integrals has a subsequence that is stronglym-orthonormal inL2(m∗), wherem∗is anotherℓ2-valued vector measure that satisfiesL2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to anℓ2-valued positive vector measure contains a lattice copy ofℓ2.
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