Abstract
Recall that a normed linear space X is a Banach space if and only if given any absolutely summable series in ∑n x n in X, lim n ∑ n k-1 x k exists. Of course, in case X is a Banach space, this gives the following implication for a series ∑ n x n : if ∑ n ∥x n ∥ < ∞, then ∑ n x n is unconditionally convergent; that is, ∑ n x π(n) converges for each permutation π of the natural numbers.
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Bibliography
Diestel, J. 1980. Measure theory in action: absolutely 2-summing operators. Proceedings of the Conference on Measure Theory and its Applications, Northern Illinois University, DeKalb, I11.
Dvoretsky, A, 1961. Some results on convex bodies and Banach spaces. Proc. Internat. Sympos. Linear Spaces, Jerusalem: Hebrew University, pp. 123 – 160.
Dvoretsky, A. and Rogers, C. A. 1950. Absolute and unconditional convergence in normed spaces. Proc. Nat. Acad. Sci. USA, 36, 192 – 197.
Figiel, T., Lindenstrauss, J., and Milman, V. 1977. The dimension of almost spherical sections of convex sets. Acta Math., 139, 53 – 94.
Grothendieck, A. 1956. Resumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paolo, 8, 1 – 79.
Maurey, B. and Pelczynski, A. 1976. A criterion for compositions of (p, q)-absolutely summing operators to be compact. Studia Math., 54, 291 – 300.
Pelczynski, A. 1976. Banach Spaces of Analytic Functions and Absolutely Summing Operators. CBMS Regional Conference Series in Mathematics, volume 30.
Pietsch, A. 1967. Absolute p-summierende Abbildungen in ncrmierten Räumen, Studia Math., 28, 333 – 353.
Stegall, C. and Retherford, J. R. 1972. Fully nuclear and completely nuclear operators with applications to ℒ1-and ℒ∞-spaces. Trans. Amer. Math. Soc., 163, 457 – 492.
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© 1984 Springer-Verlag New York, Inc.
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Diestel, J. (1984). The Dvoretsky-Rogers Theorem. In: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5200-9_6
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DOI: https://doi.org/10.1007/978-1-4612-5200-9_6
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