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  • Author or Editor: Ronglu Li x
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Suppose X is a locally convex space, Y is a topological vector space and λ(X)βY is the β-dual of some X valued sequence space λ(X). When λ(X) is c 0(X) or l (X), we have found the largest M ⊂ 2λ(X) for which (A j) ∈ λ(X)βY if and only if Σ j=1 A j(x j) converges uniformly with respect to (x j) in any MM. Also, a remark is given when λ(X) is l p(X) for 0 < p < + ∞.

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The concept of absolute convergence for series is generalized to locally convex spaces and an invariant theorem for absolutely convergent series in duality is established: when a locally convex space X is weakly sequentially complete, an admissible topology which is strictly stronger than the weak topology on X in the dual pair ( X,X′ ) is given such that it has the same absolutely convergent series as the weak topology in X .

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The key point of subseries convergence is discovered and the strongest Orlicz-Pettis-type result is established.

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