In this paper, we prove that if (Xa)a\in I are Banach spaces and X=(\oplusa\in IXa)1 their l1-sum, then the space X has the Schur property iff each factor Xa has it.
📌 3 Key Takeaways
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Objective & Hypothesis:It is a known fact that \ell1, the dual space of the null sequences c0, has the Schur property, that is, weakly convergent sequences in \ell1 are norm convergent.
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Key Findings & Data:It is a known fact that \ell1, the dual space of the null sequences c0, has the Schur property, that is, weakly convergent sequences in \ell1 are norm convergent.
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Conclusion & Impact:In this paper, we prove that if (Xa)a\in I are Banach spaces and X=(\oplusa\in IXa)1 their l1-sum, then the space X has the Schur property iff each factor Xa has it.
It is a known fact that \ell1, the dual space of the null sequences c0, has the Schur property, that is, weakly convergent sequences in \ell1 are norm convergent. In this paper, we prove that if (Xa)a\in I are Banach spaces and X=(\oplusa\in IXa)1 their l1-sum, then the space X has the Schur property iff each factor Xa has it.
Özet
It is a known fact that \ell1, the dual space of the null sequences c0, has the Schur property, that is, weakly convergent sequences in \ell1 are norm convergent. In this paper, we prove that if (Xa)a\in I are Banach spaces and X=(\oplusa\in IXa)1 their l1-sum, then the space X has the Schur property iff each factor Xa has it.
TANBAY, B. (2022). Direct Sums and the Schur Property. Turkish Journal of Mathematics, Cilt: 22 - Sayı: 3, 349–354. https://ajindex.com/articles/609000