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Perfectly Homogeneous Bases in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

P. G. Casazza  and
Bor-Luh Lin
P. G. Casazza
Affiliation:
Department of Mathematics, The University of Alabama at Huntsville Huntsville, Alabama 35807
Bor-Luh Lin
Affiliation:
Department of Mathematics 101 MacLean Hall, The University of Iowa Iowa City, Iowa 52242
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A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 137 - 140
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Altshuler, Z., Casazza, P. G. and Lin, Bor-Luh, On symmetric sequences in Lorentz sequence spaces. Israel J. Math, (to appear).Google Scholar
2. Casazza, P. G. and Lin, Bor-Luh, On symmetric basic sequences in Lorentz sequence spaces II. (Submitted).Google Scholar
3. Singer, I., Bases in Banach spaces I. Springer-Verlag 1970.Google Scholar
4. Zippin, M., On perfectly homogeneous bases in Banach spaces. Israel J. Math. 4 (1966), 265-272.Google Scholar