Abstract Duality Pairs in Analysis

The following sections are included:

• Dedication
• Contents
• Preface

The notion of abstract duality pairs or abstract triples is a generalization of the notion of dual pairs of vector spaces which is utilized in the development of locally convex spaces. A pair of (real) vector spaces E, F is a dual pair if there exists a bilinear map

In this chapter we study versions of the Orlicz–Pettis Theorem for subseries convergent series in abstract triples and use these results to establish versions of the theorem in various settings. We also give applications of the results to various topics in analysis. A series ${\displaystyle {\sum }_{j=1}^{\infty }{x}_j}$ in a topological Abelian group (X, τ) is τ subseries convergent if for every subsequence {n_j} the subseries ${\displaystyle {\sum }_{j=1}^{\infty }{x}_{n_j}}$ is τ convergent in X. 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 ([Or],[Pe]). The theorem has important applications to many areas in the integration of vector valued functions and vector valued measures. In particular, Pettis used the theorem to establish the countable additivity of the Pettis integral which he defined. The theorem has been extended to locally convex spaces and many other situations including topological groups. See [DU],[K1],[FL] for a discussion of the history of the subject. We refer to any result which asserts that a series which is subseries convergent in some weak topology is subseries convergent in a stronger topology as an Orlicz–Pettis Theorem…

A series ${\displaystyle {\sum }_j{x}_j}$ in a TVS is subseries convergent if the series

As noted earlier in Chapter 3 a series ${\displaystyle \sum {}_j}{x}_j$ in a TVS E is subseries convergent iff the series

The Uniform Boundedness Principle (UBP) was one of the early abstract results in the history of functional analysis and has found applications in many areas of analysis (see [Di], [Sw8],[Sw9] for the history). The classic version of the theorem asserts that if X is a Banach space, Y is a normed space and Γ is a subset of the space L(X, Y ) of continuous linear operators from X into Y which is pointwise bounded on X, then Γ satisfies

As noted earlier there are two possible interpretations for the conclusion of the classical Uniform Boundedness Principle (UBP). If E is a Banach space, G is a normed space and Γ is a subset of the continuous linear operators from E into G which is pointwise bounded on E, then

In this chapter we consider continuity results for biadditive and bilinear operators. These operators fit nicely into the framework of abstract triples. Let E, G be Hausdorff Abelian topological groups and F a set. Let b : E × F → G be such that b(·, y) : E → G is additive and continuous for every y ∈ F. We consider the triple

In the paper [NS], D. Noll and W. Stadler introduced an abstract notion of sectional operators modeled on the natural sectional projections of sequence spaces. They established a number of results for such sectional operators and gave applications to scalar sequence and function spaces. In [ZCL] Zheng, Cui and Li gave a generalization of sectional operators to abstract triples and established several uniform convergence results along with applications…

Let E, F be (real) vector spaces and G a topological vector space and assume that there is a bilinear map b : E × F → G. We consider the abstract triple

The following sections are included:

• Appendix A: Topology
• Appendix B: Sequence Spaces
• Appendix C: Boundedness Criterion
• Appendix D: Drewnowski
• Appendix E: Antosik–Mikusinski Matrix Theorems

The following sections are included:

• References
• Index