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On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

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Zink R.

Colloquium Mathematicum

2002

tom 92

nr 1

97-110

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 ≤ p < ∞. 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 ≤ p < ∞, is given, and a class of examples is developed in order to demonstrate the sorts of difficulties that may be encountered.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Zink R.

1 2002

Wyniki wyszukiwania

On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems