Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T17:00:22.407Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Unconditional Bases

Published online by Cambridge University Press:  20 November 2018

J. R. Holub  and
J. R. Retherford
J. R. Holub
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
J. R. Retherford
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that

1.1

convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by

1.2

where then fi(xj) = δij and fi∊X* for each positive integer i.

A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 3 , September 1972 , pp. 369 - 372
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Banach, S., Théporie des opérations linéaires, Monografje Matematyczne, Warszawa, 1932.Google Scholar
2. Day, M. M., Normed linear space, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1958.Google Scholar
3. Gelbaum, B. R., Notes on Banach spaces and Bases, An. Acad. Brasil Ci. 30 (1958), 29-36.Google Scholar
4. Grinblyum, M., Sur la théorie des systèmes biorthogonaux, C. R., Dokl. Akad. Nauk SSSR (NS), (1947), 287-290.Google Scholar
5. Orlicz, W., Uber unbedingte Konvergenz in Funktionenraumen I, Studia Math. 4 (1933), 33-37.Google Scholar
6. Orlicz, W., Beitrage zur Theorie der Orthogalentwicklungen, II, Studia Math. 8 (1929), 241-255.Google Scholar
7. Pelczynski, A., Universal Bases, Studia Math. 32 (1969), 247-268.Google Scholar
8. Pelczynski, A. and Kadec, M. I., Bases, lacunary sequences and complemented subspaces in the spaces Lp, Studia Math. 21 (1962), 161-176.Google Scholar
9. Singer, I., Bases in Banach spaces I, Springer-Verlag, Heidelberg, 1970.Google Scholar
10. Veic, B. E., Some characteristic properties of unconditional bases, Dokl. Acad. Nauk. SSS. 155 (1964), 509-512.Google Scholar
11. Veic, B. E., Translation of 10, Soviet Math. Dokl. (3) 5 (1964), 436-439.Google Scholar
12. Veic, B. E., On some properties of unconditional convergence bases, Uspehi Mat. Nauk 17 (1962), 135-142.Google Scholar