by Charles Swartz (New Mexico State University, USA)

Table of Contents (179k)
Preface (78k)
Chapter 1: Introduction (102k)

If λ is a space of scalar-valued sequences, then a series ∑_j x_j in a topological vector space X is λ-multiplier convergent if the series ∑_j=1^∞ t_jx_j converges in X for every {t_j} ελ. This monograph studies properties of such series and gives applications to topics in locally convex spaces and vector-valued measures. A number of versions of the Orlicz–Pettis theorem are derived for multiplier convergent series with respect to various locally convex topologies. Variants of the classical Hahn–Schur theorem on the equivalence of weak and norm convergent series in ι¹ are also developed for multiplier convergent series. Finally, the notion of multiplier convergent series is extended to operator-valued series and vector-valued multipliers.

• Basic Properties of Multiplier Convergent Series
• Applications of Multiplier Convergent Series
• The Orlicz–Pettis Theorem
• Orlicz–Pettis Theorems for Strong Topology
• Orlicz–Pettis Theorems for Linear Operators
• The Hahn–Schur Theorem
• Spaces of Multiplier Convergent Series and Multipliers
• The Antosik Interchange Theorem
• Automatic Continuity of Matrix Mappings
• Operator-Valued Series and Vector-Valued Multipliers
• Orlicz–Pettis Theorems for Operator-Valued Series
• Hahn–Schur Theorems for Operator-Valued Series
• Automatic Continuity for Operator-Valued Matrices