On the Gram–Schmidt orthonormalizatons of subsystems of Schauder systems
In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram–Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on $[0,1]$ is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram–Schmidt orthonormalization of any Schauder system is a Schauder basis not only for $C[0,1]$, but also for each of the spaces $L^p [0,1]$, $1 \leq p < \infty $. Although perhaps not probable, the latter result would seem to be a plausible one, since a Schauder system is closed, in the classical sense, in each of the $L^p$-spaces. This closure condition is not a sufficient one, however, since a great variety of subsystems can be removed from a Schauder system without losing the closure property, but it is not always the case that the orthonormalizations of the residual systems thus obtained are Schauder bases for each of the $L^p$-spaces. In the present work, this situation is examined in some detail; a characterization of those subsystems whose orthonormalizations are Schauder bases for each of the spaces $L^p [0,1]$, $1 \leq p < \infty $, is given, and a class of examples is developed in order to demonstrate the sorts of difficulties that may be encountered.