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 , 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 -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 -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 , 1 ≤ p < ∞, is given, and a class of examples is developed in order to demonstrate the sorts of difficulties that may be encountered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-9,
author = {Robert E. Zink},
title = {On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {97-110},
zbl = {0992.42010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-9}
}
Robert E. Zink. On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems. Colloquium Mathematicae, Tome 91 (2002) pp. 97-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-9/