Orthogonal systems in \(L^{2}\) spaces of a vector measure (Q2873175)

Let \((\Omega,\Sigma)\) be a measurable space and consider a Banach space-valued vector measure \(m:\Sigma \to X\). This paper deals with the geometric structure of the space \(L^2(m)\) of square integrable functions with respect to the vector measure \(m\). These spaces are, in general, not Hilbert spaces. In fact, they represent the class of all order continuous \(2\)-convex Banach function spaces with a weak unit. However, there are some orthogonality properties of the Hilbert function spaces that still remain true for these spaces. In particular, the notion of vector-valued orthogonality makes sense, and provides for sequences of functions satisfying this property results similar to the ones that hold for Hilbert spaces. This notion is more restrictive than the usual orthogonality in the case of Hilbert spaces, and must be understood as some sort of uniform orthogonality with respect to a class of scalar products defined by different scalar measures. A lot of applications of these structures have been published in recent years.NEWLINENEWLINEThe main result of the paper provides a constructive procedure for creating vector measure orthogonal systems. The paper contains a lot of examples -- some of them coming from classical analysis -- that make it easy to read and the fundamental purposes easy to understand. However, it uses some deep classical results of functional analysis, like the Kadec-Pelczyński disjointification method for order continuous Banach function spaces.