Weak orthogonal sequences in \(L^2\) of a vector measure and the Menchoff-Rademacher theorem (Q2428776)

Let \(L_2({\mathbf m})\) denote the Banach space of 2-integrable scalar valued functions with respect to a positive, countable Banach lattice valued measure \({\mathbf m}\). Although they are not Hilbert spaces in general, the notion of orthogonal sequences still appears in at least two natural ways, namely strong and weak orthogonality. Let \(X\) be the Banach lattice and for each positive \(x'\in X'\) look at the sequence \(S\) in \(L_2(\langle {\mathbf m},x' \rangle)\). Weak orthogonality means that for some positive functional \(x'\), \(S\) is orthogonal in \(L_2(\langle {\mathbf m},x' \rangle)\) and \(\int f_i^2\;d\langle {\mathbf m},x' \rangle >0\) for all \(f_i\in S\). The perhaps most important result of the paper under review is, given a sequence \(S\subset L_2({\mathbf m})\), a characterization of the positive real sequences \((\epsilon_i)\) with the property that there indeed exists a positive \(x'\) such that \(S\) is orthogonal in \(L_2(\langle {\mathbf m},x' \rangle)\) and \(\int f_i^2\;d\langle {\mathbf m},x' \rangle =\epsilon_i\) for all \(f_i\in S\). The paper is divided into 5 sections, where Section 1 is the introduction. Section 2 provides the reader with necessary background, while the above result is the core of Section 3. In Section 4 the authors start off by proving a theorem on when to obtain \(\langle {\mathbf m},x' \rangle\) -- a.e. convergence for a series \(\sum_n a_n f_n\) where \((f_n)\) is orthogonal with respect to \(x'\) and the \(a_n\)'s are real numbers. Next, these two theorems are combined to give the Corollaries 4.4 and 4.5. In the final Section 5, results are applied to some \(c_0\)-sums of \(L_2(\mu)\)-spaces and it is explained why these particular sums are of special interest.