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 (xn) and (yn), respectively. If each of the bases (xn) and (yn) 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have