Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

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Series in non-metrizable spaces. Theorem of Bonet and Defant. 2000. Let � uk be a convergent series in the nuclear (DF)-space E = indnEn (then it converges in a Banach step E n(0)). The following holds: (a) S( � uk) = � ∞ 1 uk + Γ ⊥ loc (� uk), where Γ ⊥ loc( � uk) := � n≥n(0) is a subspace of E. {x ∈ En | 〈x, x ′ 〉 = 0 ∀x ′ ∈ E ′ n with ∞� |〈uk, x ′ 〉| < ∞} (b) If E is not isomorphic to the direct sum ϕ of copies of R, then there is a convergent series in E whose set of sums is a non-closed subspace of E. José Bonet Rearrangement of series. The theorem of Levy-Steiniz. 1
Series in non-metrizable spaces. Theorem of Bonet and Defant. 2000. Let E = indnEn be a complete (DF)-space such that every convergent sequence in E converges in one of the Banach spaces En. If we have S( � uk) = ∞� 1 uk + Γ ⊥ loc( � uk) for every convergent series � uk in E, then the space E is nuclear. José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
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