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Vector measure orthonormal functions and best approximation for the 4-norm. (English) Zbl 1036.46031

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.

MSC:

46G10 Vector-valued measures and integration
46B45 Banach sequence spaces
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