On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink. "On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems." Colloquium Mathematicae 92.1 (2002): 97-110. <http://eudml.org/doc/283458>.

@article{RobertE2002,
abstract = {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.},
author = {Robert E. Zink},
journal = {Colloquium Mathematicae},
keywords = {Franklin system; Gram-Schmidt orthonormalization; Schauder basis; Schauder system; -spaces},
language = {eng},
number = {1},
pages = {97-110},
title = {On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems},
url = {http://eudml.org/doc/283458},
volume = {92},
year = {2002},
}

TY - JOUR
AU - Robert E. Zink
TI - On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems
JO - Colloquium Mathematicae
PY - 2002
VL - 92
IS - 1
SP - 97
EP - 110
AB - 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.
LA - eng
KW - Franklin system; Gram-Schmidt orthonormalization; Schauder basis; Schauder system; -spaces
UR - http://eudml.org/doc/283458
ER -