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The Convergence of Series for Various Choices of Sign in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

James Shirey
James Shirey*
Affiliation:
Ohio University, Athens, Ohio
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1. Let (xn, Xn) denote a basis for a Banach space (X, ∥ • ∥) of measurable functions in (0, 1).

It is shown in [2] and [9] that the equivalence of the norms

and ∥ • ∥ is equivalent to the unconditionality of the basis (xn, Xn). In [8] a weaker relationship between these norms is exploited to establish the existence of an element of L1(E) for each E ⊂ (0, 1), |£| > 0, whose Haar series expansion is conditionally convergent in the norm of L\(E).

In this note, a Lemma of Orlicz [7] is generalized to provide a relationship between , and the changes in sign that are tolerated in without disruption of norm convergence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 475 - 480
Copyright
Copyright © Canadian Mathematical Society 1975

References

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