Abstract
We prove the following Cantor–Bernstein type theorem, which applies well to the class of symmetric sequence spaces studied earlier by Altshuler, Casazza, and Lin: Let X and Y be Banach spaces having symmetric bases (x n ) and (y n ), respectively. If each of the bases (x n ) and (y n ) is equivalent to a basic sequence generated by one vector of the other, then the spaces X and Y are isomorphic. As a consequence, we obtain the strong equivalence that two Lorentz sequence spaces have the same linear dimension if and only if they are isomorphic.
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This is part of the author’s doctoral dissertation under the supervision of Prof. Marek Wójtowicz (Advisor) and Prof. Carlos E. Finol (Co-Advisor).
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González, M.J. Cantor–Bernstein theorems for certain symmetric bases in Banach spaces. Arch. Math. 105, 425–433 (2015). https://doi.org/10.1007/s00013-015-0814-x
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DOI: https://doi.org/10.1007/s00013-015-0814-x