The Diestel-Faires Theorem on Series

Swartz, Charles

Tatra Mountains Mathematical Publications, Tome 45 (2010), / Harvested from Mathematical Institute

Résumé

We give a proof of an Orlicz-Pettis Theorem of Diestel and Faires on weak*subseries convergent series in the dual of a Banach space using an elementarytheorem on real valued matrices.\bigskipThe Orlicz-Pettis Theorem on subseries convergence has proven to be one of themost useful theorems in functional analysis with applications to Banach spacetheory, vector measures and vector integration. The version of the theorem fornormed spaces asserts that a series which is subseries convergent in the weaktopology is subseries convergent in the norm topology (for the history of theOrlicz-Pettis Theorem, see [FL],[DU],[Ka]). Simple examples show that theanalogue of the Orlicz-Pettis Theorem fails for the weak* topology of dualspaces (see Example 1), and, in fact, Diestel and Faires have shown that aBanach space $X$ has the property that series in the dual $X^{\prime}$ areweak* subseries convergent iff they are norm subseries convergent$\Longleftrightarrow$ the space $X^{\prime}$ contains no subspace isomorphicto $l^{\infty}$. This result of Diestel/Faires is actually a corollary of amuch more general result concerning vector valued measures. There have been anumber of additional proofs of the Diestel/Faires result, but all of theproofs, including the original, use non-trivial properties of vector measures.For example, the proof in Diestel/Uhl ([DU]) uses a lemma of Rosenthal onvector measures and the proof in [Sw2] uses a lemma of Drewnowski on finitelyadditive set functions. Since the statement of the Diestel/Faires result forseries involves only series, it would seem to be desirable to give a proofwhich only involves basic properties of series and does not invoke propertiesof vector valued measures. In this brief note we will show that a simpletheorem about real valued infinite matrices given in [AS] can be employed togive a proof of the Diestel/Faires result which involves only basic propertiesof series in normed spaces (actually we consider only one part of theDiestel/Faires result).First, we give an example showing a straightforward analogue of theOrlicz-Pettis Theorem fails for the weak* topology

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Publié le : 2010-01-01
DOI : https://doi.org/10.2478/tatra.v46i0.82