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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Basic sequences and subspaces in Lorentz sequence spaces without local convexity


Author: Nicolae Popa
Journal: Trans. Amer. Math. Soc. 263 (1981), 431-456
MSC: Primary 46A45; Secondary 46A10
DOI: https://doi.org/10.1090/S0002-9947-1981-0594418-7
MathSciNet review: 594418
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Abstract: After some preliminary results $ (\S1)$, we give in $ \S2$ another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of $ {l_p}$, $ 0 < p < 1$.

Using this result we prove in §3 the unicity of certain bounded symmetric block bases of the subspaces of the Lorentz sequence spaces $ d(w,p)$, $ 0 < p < 1$. In $ \S4$ we show that every infinite dimensional subspace of $ d(w,p)$ contains a subspace linearly homeomorphic to $ {l_p}$, $ 0 < p < 1$.

Unlike the case $ p \geqslant 1$ there are subspaces of $ d(w,p)$, $ 0 < p < 1$, which contain no complemented subspaces of $ d(w,p)$ linearly homeomorphic to $ {l_p}$. In fact there are spaces $ d(w,p)$, $ 0 < p < 1$, which contain no complemented subspaces linearly homeomorphic to $ {l_p}$. We conjecture that this is true for every $ d(w,p)$, $ 0 < p < 1$. The answer to the previous question seems to be important: for example we can prove that a positive complemented sublattice $ E$ of $ d(w,p)$, $ 0 < p < 1$, with a symmetric basis is linearly homeomorphic either to $ {l_p}$ or to $ d(w,p)$; consequently, a positive answer to this question implies that $ E$ is linearly homeomorphic to $ d(w,p)$. In $ \S5$ we are able to characterise the sublattices of $ d(w,p)$, $ p = {k^{ - 1}}$ (however under a supplementary restriction concerning the sequence $ ({w_n})_{n = 1}^\infty )$, which are positive and contractive complemented, as being the order ideals of $ d(w,p)$.

Finally, in $ \S6$, we characterise the Mackey completion of $ d(w,p)$ also in the case $ p = {k^{ - 1}}$, $ k \in {\mathbf{N}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0594418-7
Keywords: $ p$-Banach spaces, symmetric bases, complemented subspaces
Article copyright: © Copyright 1981 American Mathematical Society