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On Cesaro summable sequences of continuous functions

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  26 February 2010

Sophocles Mercourakis
Sophocles Mercourakis
Affiliation:
University of Athens, Department of Mathematics, Panepistemiopolis, 15784 Athens, Greece.
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Abstract

A new type of convergence (called uniformly pointwise convergence) for a sequence of scalar valued functions is introduced. If (fn) is a uniformly bounded sequence of functions in l∞(Γ), it is proved that:

(i) (fn) converges uniformly pointwise on Γ to some function f if, and only if, every subsequence of (fn) is Cesaro summable in l∞(Γ); and

(ii) there exists a subsequence (f′n) of (fn) such that either (f′n) converges uniformly pointwise on Γ to some f or no subsequence of (f′n) is Cesaro-summable in l∞(Γ).

Applications of the above results in Banach space theory are given.

MSC classification

Secondary: 46A19: Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${bf R}$, etc.)
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 87 - 104
Copyright
Copyright © University College London 1995

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