On the Gram–Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink¹

¹ Department of Mathematics Purdue University West Lafayette, IN 47907, U.S.A.

Colloquium Mathematicum, Tome 92 (2002) no. 1, pp. 97-110.

Résumé

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 $L^p[0,1]$, $1\le p\mathrm{\infty }$. 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 $L^p[0,1]$, $1\le p\mathrm{\infty }$, is given, and a class of examples is developed in order to demonstrate the sorts of difficulties that may be encountered.

Détail

DOI : 10.4064/cm92-1-9

Mots-clés : earliest monographs involve notion schauder basis franklin showed gram schmidt orthonormalization certain schauder basis banach space functions continuous again schauder basis space subsequently ciesielski observed gram schmidt orthonormalization schauder system schauder basis only each spaces leq infty although perhaps probable latter result would seem plausible since schauder system closed classical sense each p spaces closure condition sufficient however since great variety subsystems removed schauder system without losing closure property always the orthonormalizations residual systems obtained schauder bases each p spaces present work situation examined detail characterization those subsystems whose orthonormalizations schauder bases each spaces leq infty given class examples developed order demonstrate sorts difficulties may encountered