The Diestel-Faires Theorem on Series
Abstract
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 elementary
theorem on real valued matrices.
\bigskip
The Orlicz-Pettis Theorem on subseries convergence has proven to be one of the
most useful theorems in functional analysis with applications to Banach space
theory, vector measures and vector integration. The version of the theorem for
normed spaces asserts that a series which is subseries convergent in the weak
topology is subseries convergent in the norm topology (for the history of the
Orlicz-Pettis Theorem, see [FL],[DU],[Ka]). Simple examples show that the
analogue of the Orlicz-Pettis Theorem fails for the weak* topology of dual
spaces (see Example 1), and, in fact, Diestel and Faires have shown that a
Banach space $X$ has the property that series in the dual $X^{\prime}$ are
weak* subseries convergent iff they are norm subseries convergent
$\Longleftrightarrow$ the space $X^{\prime}$ contains no subspace isomorphic
to $l^{\infty}$. This result of Diestel/Faires is actually a corollary of a
much more general result concerning vector valued measures. There have been a
number of additional proofs of the Diestel/Faires result, but all of the
proofs, including the original, use non-trivial properties of vector measures.
For example, the proof in Diestel/Uhl ([DU]) uses a lemma of Rosenthal on
vector measures and the proof in [Sw2] uses a lemma of Drewnowski on finitely
additive set functions. Since the statement of the Diestel/Faires result for
series involves only series, it would seem to be desirable to give a proof
which only involves basic properties of series and does not invoke properties
of vector valued measures. In this brief note we will show that a simple
theorem about real valued infinite matrices given in [AS] can be employed to
give a proof of the Diestel/Faires result which involves only basic properties
of series in normed spaces (actually we consider only one part of the
Diestel/Faires result).
First, we give an example showing a straightforward analogue of the
Orlicz-Pettis Theorem fails for the weak* topology
subseries convergent series in the dual of a Banach space using an elementary
theorem on real valued matrices.
\bigskip
The Orlicz-Pettis Theorem on subseries convergence has proven to be one of the
most useful theorems in functional analysis with applications to Banach space
theory, vector measures and vector integration. The version of the theorem for
normed spaces asserts that a series which is subseries convergent in the weak
topology is subseries convergent in the norm topology (for the history of the
Orlicz-Pettis Theorem, see [FL],[DU],[Ka]). Simple examples show that the
analogue of the Orlicz-Pettis Theorem fails for the weak* topology of dual
spaces (see Example 1), and, in fact, Diestel and Faires have shown that a
Banach space $X$ has the property that series in the dual $X^{\prime}$ are
weak* subseries convergent iff they are norm subseries convergent
$\Longleftrightarrow$ the space $X^{\prime}$ contains no subspace isomorphic
to $l^{\infty}$. This result of Diestel/Faires is actually a corollary of a
much more general result concerning vector valued measures. There have been a
number of additional proofs of the Diestel/Faires result, but all of the
proofs, including the original, use non-trivial properties of vector measures.
For example, the proof in Diestel/Uhl ([DU]) uses a lemma of Rosenthal on
vector measures and the proof in [Sw2] uses a lemma of Drewnowski on finitely
additive set functions. Since the statement of the Diestel/Faires result for
series involves only series, it would seem to be desirable to give a proof
which only involves basic properties of series and does not invoke properties
of vector valued measures. In this brief note we will show that a simple
theorem about real valued infinite matrices given in [AS] can be employed to
give a proof of the Diestel/Faires result which involves only basic properties
of series in normed spaces (actually we consider only one part of the
Diestel/Faires result).
First, we give an example showing a straightforward analogue of the
Orlicz-Pettis Theorem fails for the weak* topology
Full Text:
PDFDOI: https://doi.org/10.2478/tatra.v46i0.82