Multiplier Convergent Series

The following sections are included:

• Preface

• Contents

One of the most interesting and useful theorems in the early history of functional analysis is a result now known as the Orlicz-Pettis Theorem. The result was originally proven by Orlicz for weakly sequentially complete normed spaces although the result in full generality for normed spaces was known by the Polish mathematicians and appears in Banach's book ([Or], [Ba]). The first version of the theorem available in English was proven by Pettis and was used to treat topics in vector valued measures and vector valued integrals ([Pe]; see [Ka3] and [FL] for discussions of the history of the theorem). If X is a topological vector space (TVS), a series ∑_jx_j in X is subseries convergent in X if the subseries converges in X for every subsequence {n_j}. The Orlicz-Pettis Theorem for normed spaces states that if the series ∑_jx_j is subseries convergent in the weak topology of the space, then the series is actually subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was extended to locally convex spaces by McArthur ([Mc]). If σ is any subset of ℕ and χ_σ is the characteristic function of σ, then a series ∑_jx_j in a TVS X is subseries convergent iff the series converges in X for every σ ⊂ ℕ. Thus, if m₀ = span{χ_σ : σ ⊂ ℕ}, the sequence space of real valued sequences with finite range, a series ∑_jx_j in a TVS X is subseries convergent iff the series converges for every t = {t_j} ∈ m₀. To obtain a generalization of the notion of subseries convergence, we may replace the space m₀ by a general vector space λ of real valued sequences. If λ is a vector space of real valued sequences and {x_j} is a sequence in the TVS X, the (formal) series ∑_jx_j is said to be λ multiplier convergent if the series converges in X for every t = {t_j} ∈ λ the elements t ∈ λ are called multipliers. This suggests that generalizations of the Orlicz-Pettis Theorem might be obtained by replacing subseries convergent series by λ multiplier convergent series for certain sequence spaces λ. We show that such generalizations are possible in Chapters 4, 5, and 6…

In this chapter we will develop the basic properties of multiplier convergent series.

In what follows λ will denote a scalar sequence space which contains the subspace c₀₀ of sequences which are eventually 0 and Λ ⊂ λ will denote a subset while X will denote a Hausdorff topological vector space (TVS)…

In this chapter we will give several applications of multiplier convergent series to various topics in locally convex spaces and vector valued measures. As before, throughout this chapter λ will denote a sequence space containing c₀₀, the space of sequences which are eventually 0 and X will denote a (Hausdorff) LCTVS. We begin by establishing a generalization of a result of G. Bennett ([Be])…

As noted earlier the classical version of the Orlicz-Pettis Theorem for normed spaces asserts that a series in a normed space which is subseries convergent in the weak topology of the space is subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was originally established by Orlicz for weakly sequentially complete spaces but was evidently known in full generality by the Polish mathematicians as it appears as a statement in Banach's book ([Ba]). The first version available in English was established by Pettis in [Pe] where it was used to treat topics in vector valued integration — the Pettis integral. The theorem was extended to locally convex spaces by McArthur ([Mc]). For historical discussions of the theorem see [Ka3], [DU], or [FL]. Since a series is subseries convergent iff the series is m₀ multiplier convergent, it is natural to ask what sequence spaces λ have the property that series which are λ multiplier convergent in the weak topology are λ multiplier convergent in some stronger topology such as the Mackey topology. We will refer to such results as Orlicz-Pettis Theorems…

In this chapter we consider Orlicz-Pettis Theorems for the strong topology. As the following example shows, in general, an Orlicz-Pettis Theorem does not hold for the strong topology. Recall that if X, X′ is a pair of vector spaces in duality, the strong topology β(X,X′) is the polar topology of uniform convergence on the family of σ(X′,X) bounded subsets of X′ (Appendix A, Example A.2). As before, throughout this chapter λ will denote a scalar sequence space which contains c₀₀, the space of sequences which are eventually 0…

In this chapter we consider multiplier convergent series of continuous linear operators and establish Orlicz-Pettis Theorems for such series. Throughout this chapter let X and Y ≠ {0} be LCTVS and L(X, Y) the space of continuous linear operators from X into Y. We first describe the topologies on L(X, Y) which will be considered…

In this chapter we establish vector versions of the classical Hahn-Schur Theorem for multiplier convergent series. For later reference, we first give a statement of one version of the Schur Theorem for absolutely convergent scalar valued series…

In this chapter we consider topological properties of the space of λ multiplier convergent series. Throughout this chapter let λ be a sequence space containing c₀₀ and let X be an LCTVS. Recall that λ^βX is the space of all X valued λ convergent series. If x = {x_j} ∈ λ^βX and t = {t_j} ∈ λ, we write We define a locally convex topology on λ^βX induced by X and λ when λ is a K-space…

A problem often encountered in analysis is the interchange of two limiting processes. For example, the Lebesgue Dominated Convergence Theorem gives sufficient conditions to interchange the pointwise limit of a sequence of integrable functions with the Lebesgue integral, i.e., to take the “limit under the integral sign”. In this chapter we consider sufficient conditions for the equality of two iterated series. For real valued series one of the most useful criterion for interchanging the limit of an iterated series is the absolute convergence of the iterated series. However, absolute convergence for series with values in an LCTVS is a very strong condition and is, therefore, not appropriate. Antosik has given a sufficient condition involving subseries convergence of an iterated series with values in a topological group which has proven to be useful in a number of applications ([A]). We begin this chapter with a presentation of Antosik's result for series with values in a TVS. We then give generalizations of Antosik's result to multiplier convergent series…

In this chapter we will consider the automatic continuity and boundedness of matrix mappings between sequence spaces which are K-spaces. Let λ and μ be sequence spaces which contain c₀₀, and let A = [a_ij] be an infinite matrix with real entries. The matrix A maps λ into μ, denoted by A : λ → μ, if for each t ∈ λ, the series converge for every i and the sequence belongs to μ; we write . Thus, for a matrix A to map λ into μ, it is necessary for the rows of A to be λ multiplier convergent. We will consider the continuity and boundednessof a matrix A : λ → μ with respect to various topologies on λ and μ. Such results are often referred to as automatic continuity (boundedness) results since there are no apriori assumptions on the matrix A. The first such automatic continuity result for matrix mappings is due to Hellinger and Toeplitz who showed that a matrix A which maps l² into itself is automatically continuous ([HT]). Another general automatic continuity which follows from the Closed Graph Theorem is that any matrix A mapping an FK-space λ into another FK-space μ is continuous [recall that a sequence space is an FK-space if it is a K-space under a complete metrizable vectortopology (Appendix B)]. We now give a proof of a generalization of this result…

In this chapter we will consider operator valued series but allow the space of multipliers to be vector valued with values in the domain space of the operators. Throughout this chapter let X, Y be LCTVS and L(X, Y) the space of all continuous linear operators from X into Y. Let E be a vector valued sequence space with values in X which contains c₀₀(X), the space of all X valued sequences which are eventually 0…

In this chapter we consider Orlicz-Pettis type theorems for operator valued series and vector valued multipliers. Throughout this chapter let X and Y be LCTVS and L(X, Y) the space of all continuous linear operators from X into Y. Let E be a vector space of X valued sequences which contains the subspace c₀₀(X) of all sequences which are eventually 0. If ∑_jT_j is a series in L(X, Y) and x = {x_j} is a multiplier, then the series has values in Y so any Orlicz-Pettis Theorem must focus on the topology of Y. We first have a straightforward result…

In this chapter we consider versions of the Hahn-Schur Theorem for series with values in the space of continuous linear operators and with multipliers which are vector valued with values in the domain of the operators. Throughout this chapter let X and Y be LCTVS, L(X, Y) the space of all continuous linear operators from X into Y, and E a vector space of X valued sequences which contains c₀₀(X), the space of all X valued sequences which are eventually 0. The analogue of the hypothesis (H) for the Hahn-Schur Theorem in Chapter 7 for E multiplier convergent series is straightforward…

In this chapter we will establish several automatic continuity results for operator valued matrices analogous to those for scalar valued matrices established in Chapter 10. We first consider the relationship between an operator matrix and its transpose which require some technical assumptions…

The following sections are included:

• Appendix A Topological Vector Spaces

• Appendix B Scalar Sequence Spaces

• Appendix C Vector Valued Sequence Spaces

• Appendix D The Antosik-Mikusinski Matrix Theorems

• Appendix E Drewnowski's Lemma

• References

• Index