Hostname: page-component-68945f75b7-w588h Total loading time: 0 Render date: 2024-09-03T11:14:00.931Z Has data issue: false hasContentIssue false
  • English
  • Français

A Class of Reflexive Symmetric Bk-Spaces

Published online by Cambridge University Press:  20 November 2018

D. J. H. Garling
D. J. H. Garling*
Affiliation:
St. John's College, Cambridge, England
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.

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A sequence space E is a BK-space (9) if it is equipped with a norm under which: first, E is a Banach space and second, each of the coordinate maps x → xi is continuous. Let ∑ be the group of all permutations of Z+ = {1, 2, 3, …}. If x ∈ ω and σ ∈ ∑, the sequence xσ is defined by (xσ)i = xσ(i)). A sequence space E is symmetric if xσ ∈ E whenever x ∈ E and σ ∈ ∑. Accounts of symmetric sequence spaces occur in (3; 7; 8).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 602 - 608
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Banach, S., Opérations linéaires (Chelsea, New York, 1955).Google Scholar
2. Bourbaki, N., Elements de mathématique, Livre V: Espaces vectoriels topologiques, Chapitres I—II, Actualités Sci. Indust., no. 1189 (Hermann, Paris, 1953).Google Scholar
3. Garling, D. J. H., On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85–105.Google Scholar
4. Grothendieck, A., Espaces vectoriels topologiques, 2e éd., mimeographed notes (Sociedade de Matemâtica de Sao Paulo, Sao Paulo, 1958).Google Scholar
5. Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc, no. 16, 1955.Google Scholar
6. Kôthe, G., Topologische lineare Ràume (Springer, Berlin, 1960).Google Scholar
7. Ruckle, W., Symmetric coordinate spaces and symmetric bases, Can. J. Math. 19 (1967), 828–838.Google Scholar
8. Sargent, W. L. C., Some sequence spaces related to the lp spaces, J. London Math. Soc. 35 (1960), 161–171.Google Scholar
9. Zeller, K., Théorie der Limitierungsverfahren (Springer, Berlin, 1958).Google Scholar
10. Zygmund, A., Trigonometric series, Vol. II (Cambridge Univ. Press, Cambridge, 1959).Google Scholar