Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:55:49.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Biorthogonality in the banach spaces ℓp(n)*

Published online by Cambridge University Press:  20 January 2009

Anthony J. Felton  and
H. P. Rogosinski
Anthony J. Felton
Affiliation:
Department of Mathematics University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, Wales
H. P. Rogosinski
Affiliation:
Department of Mathematics University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, Wales
Rights & Permissions [Opens in a new window]

Abstract

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 consider the finite-dimensional Banach spaces ℓp(n), where p>l. On these spaces there is a unique homogeneous semi-inner-product [.,.] consistent with the norm. If p≠2 this semi-inner product is not symmetric. We define a pair of vectors x and y to be biorthogonal if [x, y] = [y, x] = 0. For a given non-zero x, let τ(X) be the number of elements in a maximal linearly independent set of vectors biorthogonal to x. If p = 2 it is well-known that this number is n–1. The aim of this paper is to find τ(X) when p≠2. Our investigation shows that the situation differs from the Euclidean case in that the value of τ(X) can be either n–l or n –2. The ‘exceptional’ vectors x for which τ(x) = n –2 are characterised.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 325 - 336
Copyright
Copyright © Edinburgh Mathematical Society 1996

Footnotes

*

This research was funded by the University of Wales.

References

REFERENCES

1. Giles, J. R., Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967), 436–446.Google Scholar
2. Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.CrossRefGoogle Scholar