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Reducing the classical multipliers ℓ∞, C0 and bv0

Published online by Cambridge University Press:  20 January 2009

Stephen A. Saxon  and
William H. Ruckle
Stephen A. Saxon
Affiliation:
Department of Mathematics, University of Florida, Po Box 118000, Gainesville, FL 32611–8000, USA E-mail address: [email protected]
William H. Ruckle
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634–1907, USA E-mail address: [email protected]
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Abstract

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For R ∈ {bv0, c0, ℓ∞} a multiplier of FK spaces, the classical sectional convergence theorems permit the reduction of R to any of its dense barrelled subspaces as a simple consequence of the Closed Graph Theorem. (Cf. the Bachelis/Rosenthal reduction of R = ℓ∞ to its dense barrelled subspace m0.) A natural modern setting permits the reduction of R to any of the larger class of dense βφ subspaces. Bennett and Kalton's FK setting remarkably reduced R = ℓ∞ to any of its dense subspaces. This extreme reduction also obtains in the modern βφ setting since, surprisingly, every dense subspace of ℓ∞ is a βφ subspace. Moreover, the standard results, including the Bennett/Kalton reduction, easily follow from their βφ versions and the Closed Graph Theorem. Our two supporting papers find relevant “Non-barrelled dense βφ subspaces” and study “Generalized sectional convergence and multipliers”. Here we specialize the βφ approach to ordinary, particularly unconditional, sectional convergence.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 2 , June 1997 , pp. 345 - 352
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

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