Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An orthonormal basis for $ C[0,1]$ that is not an unconditional basis for $ L\sp p[0,1],\;1<p\not= 2$


Author: Robert E. Zink
Journal: Proc. Amer. Math. Soc. 97 (1986), 33-37
MSC: Primary 42C20; Secondary 46B15
MathSciNet review: 831382
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent article, Kazaryan has employed an orthonormal system constructed by Olevskii in order to obtain a negative answer to the following question posed by Ulyanov: Is an orthonormal basis for $ C\left[ {0,1} \right]$ necessarily an unconditional basis for each space $ {L^p}\left[ {0,1} \right],1 < p < \infty $? The elements of the Olevskii system, however, are finite linear combinations of Haar functions, and thus, most of them are not continuous on $ \left[ {0,1} \right]$. For this reason, the example is mildly unsatisfying, since one generally requires the members of a Schauder basis for a given space to belong to that space. In the present work, the author shows that this minute defect can be removed if one modifies the Olevskii system by replacing the Haar functions involved therein with corresponding members of the Franklin system.


References [Enhancements On Off] (What's this?)

  • [1] Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141–157. MR 0157182
  • [2] Z. Ciesielski, Properties of the orthonormal Franklin system. II, Studia Math. 27 (1966), 289–323. MR 0203349
  • [3] V. F. Gaposhkin, On unconditional basis in the spaces $ {L^p}\left( {p > 1} \right)$, Uspekhi Mat. Nauk 13 (1958), 179-184. (Russian)
  • [4] K. S. Kazaryan, Some questions of the theory of orthogonal series, Mat. Sb. (N.S.) 119(161) (1982), no. 2, 278–294, 304 (Russian). MR 675197
  • [5] A. M. Olevskiĭ, An orthonormal system and its applications, Mat. Sb. (N.S.) 71 (113) (1966), 297–336 (Russian). MR 0203351

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42C20, 46B15

Retrieve articles in all journals with MSC: 42C20, 46B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0831382-7
Article copyright: © Copyright 1986 American Mathematical Society