Vector measure orthonormal functions and best approximation for the 4-norm (Q1401605)

For a vector measure \(\lambda\) with values in a Hilbert space, the notions of \(\lambda\)-orthogonality for pairs of \(\lambda\)-integrable functions and \(\lambda\)-orthonormality for sequences of functions are defined. Then, it is proved that, under certain conditions, such a sequence generates a function space isometric to the sequence space \(\ell_4\) when endowed with the 4-norm. Then there follows a discussion of the relationship between \(\lambda\)-orthogonal functions and best approximation for the 4-norm.