A generalization of Drewnowski's result on the Cantor-Bernstein type theorem for a~class of nonseparable Banach spaces
| Autor: | Marek Wójtowicz, Marcos J. González |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Discrete mathematics
Pure mathematics Mathematics::Functional Analysis Functional analysis Order isomorphism General Mathematics Eberlein–Šmulian theorem Banach space Banach manifold Type (model theory) uncountable symmetric basis 46B42 Uncountable set linear dimension Levi norm Lp space Orlicz space 46B45 47B37 Mathematics discrete Banach lattice 46B26 |
| Zdroj: | Funct. Approx. Comment. Math. 50, no. 2 (2014), 283-296 |
| Popis: | Let $X_a$ denote the order continuous part of a Banach lattice $X$, and let $\Gamma$ be an uncountable set. We extend Drewnowski's theorem on the comparison of linear dimensions between Banach spaces having uncountable symmetric bases to a~class of discrete Banach lattices, the so-called $D$-spaces. We show that if $X$ and $Y$ are two $D$-spaces and there are continuous linear injections (not necessarily embeddings) from $X$ into $Y$ and vice versa, then $X$ and $Y$ are order-topologically isomorphic. In the proof we apply a theorem on the extension of an order isomorphism from $X_a$ onto $Y_a$ to an order isomorphism from $X$ onto $Y$, the classical Drewnowski's theorem, and a supplement of Troyanski's theorem on embeddings of $\ell_1(\Gamma)$ spaces into a Banach space with an uncountable symmetric basis. Our result applies to the class of Orlicz spaces $\ell_\varphi(\Gamma)$, where $\varphi$ is an Orlicz function. |
| Databáze: | OpenAIRE |
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