L'espace usuel C ne possède aucun R.U.C-système orthonormé. (The usual space C doesn't have an orthonormal R.U.C-system)
type: Item
instance of: scholarly article
title: L'espace usuel C ne possède aucun R.U.C-système orthonormé. (The usual space C doesn't have an orthonormal R.U.C-system)
zbMATH Open document ID: 0679.46009
DOI: 10.1007/BF01300064
author: P. Billard
published in: Monatshefte für Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178434
review text: It is proved that a sequence \((\phi_ n)\) in the usual Banach space C which is orthonormal and fundamental (for example the Legendre system), never is an orthonormal R.U.C-system (i.e. has the additional property that for each f in C with the expansion \(\sum a_ n\phi_ n\) we have convergence of \(\sum \pm a_ n\phi_ n\) in C for almost all choices of signs \(\pm).\) The meaning of this result is to be found in comparison with known results, especially i) there exists a basis \((\phi_ n)\) in C such that for each f in C with the expansion \(\sum a_ n\phi_ n\) we have the convergence of \(\sum \pm a_ n\phi_ n\) in C for almost all choices of signs \(\pm,\) ii) for no basis in C the words ``almost all'' in i) can be replaced by the work ``all'', iii) there exist an orthonormal basis in C (for example the Franklin system).
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 4113270
zbMATH Keywords: R.U.C-system
zbMATH Keywords: Franklin system
MaRDI profile type: Publication
cites work: Q3726769
cites work: Q3687398
cites work: Q4136031
cites work: Projections in certain Banach spaces
cites work: A Characterization of Banach Spaces Containing l 1
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description: scientific article; zbMATH DE number 4113270