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Overview
If λ is a space of scalar-valued sequences, then a series Σj xj in a topological vector space X is λ-multiplier convergent if the series Σj=1&infty; tjxj converges in X for every {tj} ∈λ. 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 l1 are also developed for multiplier convergent series. Finally, the notion of multiplier convergent series is extended to operator-valued series and vector-valued multipliers.Synopsis
If λ is a space of scalar-valued sequences, then a series Σj xj in a topological vector space X is λ-multiplier convergent if the series Σj=1&infty; tjxj converges in X for every {tj} ∈λ. 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 l1 are also developed for multiplier convergent series. Finally, the notion of multiplier convergent series is extended to operator-valued series and vector-valued multipliers.